I know Jesus says marriage ends with death, but I don't understand WHY marriage ceases to exist with death. I prefer Catholic answers to this question.
The primary end of marriage is the procreation and the education of children
Cornelius à Lapide, S.J., commentates on St. Matthew 22:30 ("For in the resurrection they shall neither marry nor be married, but shall be as the angels of God in heaven."):
there will be no need then of marriage and generation; for these things have been instituted for the perpetuation of the race and the individual, by means of children. Because the father is mortal, therefore he begets a son, that after death he may live and continue in his son. But in Heaven there shall be no death, and they shall live for ever. Marriage, therefore, and procreation of children would be without an object there. Wherefore S. Luke adds (20:36), Neither can they die any more. Appositely says S. Augustine (Quæst. Evang. in Luc. xx. 35):
Marriage is for the sake of children, children for the sake of succession, succession on account of death. Where, therefore, death is not, marriage is not.
Though marriage ceases, the bonds of charity between friends endure (St. Thomas Aquinas, Summa Theologica II-II q. 26 a. 3 "Whether the order of charity endures in heaven?"); "there seems to be the greatest friendship between husband and wife" (Summa contra Gentiles lib. 3 cap. 123 ).
Catholic teaching that marriage ceases with death
Good question (although Jesus didn't exactly say marriage ends with death), especially with the secular culture promoting the idea that "marriage is forever" (e.g. wedding ring seller: "diamond is forever"). Also, with the Catholic church assiduously protecting the indissolubility of sacramental marriage before death (if valid and consummated) and how one has to go through a possibly complicated canonical process for remarrying, why does the Catholic church teach that the marriage bond ceases with death? (see below)
Can. 1141 A marriage that is ratum et consummatum can be dissolved by no human power and by no cause, except death.
Scripture also provides the legal basis of marriage dissolution at death consistent with "Until Death Do Us Part" in the marriage vow (see the history of it in a magazine article Why We Say "Until Death Do Us Part" In Wedding Vows). Romans 7:3:
Accordingly, she will be called an adulteress if she lives with another man while her husband is alive. But if her husband dies, she is free from that law, and if she marries another man she is not an adulteress.
What will survive and will be transformed: love between spouses
To answer this question, we need to understand which aspect of earthly marriage survives to heaven according to the Catholic church. A reddit question has a good answer (emphasis mine):
Marriage is a sacrament. Sacraments are "visible signs of invisible grace" that help us encounter God and His Love in tangible ways. When we are in heaven, we will have no need of sacraments any more because we will be dwelling within God's love eternally. We won't need to receive the Eucharist, for example, since we will eternally be in God's presence.
Marriage, in particular, is a sacrament that makes God's love known on earth (to each spouse in the marriage and to the community who witnesses their love). We won't need marriage to witness to God's love in heaven, however, because we will be IN God's love already. It would be superfluous.
Of course, that's not to say that a lifetime of loving one another won't mean anything at all in heaven. The love I have for my husband, for example, will still be present.
A similar explanation is offered by Fr. Raniero Cantalamessa in his commentary on the readings for the 32nd Sunday in Ordinary Time (cycle B) which also addresses other scenarios (unhappy marriage, remarriage after first spouse's death, etc.).
A third support can be had from the conclusion of St. John Chrysostom's Letter to a Young Widow (whose husband died after only 5 years of happy marriage) that marital love will be transformed into a "far nobler kind":
Wherefore desisting from mourning and lamentation do thou hold on to the same way of life as his, yea even let it be more exact, that having speedily attained an equal standard of virtue with him, you may inhabit the same abode and be united to him again through the everlasting ages, not in this union of marriage but another far better. For this is only a bodily kind of intercourse, but then there will be a union of soul with soul more perfect, and of a far more delightful and far nobler kind.
(Source: NC Register article Is There Marriage in Heaven?)
What ceases: procreation and vow
The Catholic church teaches that the marriage covenant
is ordered by its nature to the good of the spouses and the procreation and education of offspring ... (Canon 1055)
Although in Mark 12:25 and Matthew 22:30 Jesus didn't exactly say that marriage ceases with death, Jesus explicitly said that there will be no new marriages and that couples will live like angels in heaven, implying that there will also be no procreation (like angels). Since what is "good of the spouses" has been transformed into something better (see previous section), then it makes sense that there is no more reason for the marriage covenant after death.
Also, in Romans 7:1-3 Paul implied that once a spouse has died he/she will no longer be bound by law to the other.
Non Catholic Views
It's interesting that for LDS marriage survives death: see LDS answer to the question "Do Mormons believe that marriage is eternal?"
Also, among non-Catholics there are several views of the marriage covenant that become important when considering whether Remarriage after divorce should be allowed; see a book in the Counterpoints series: Remarriage After Divorce in Today's Church: 3 Views.
Because in Heaven your bond with everyone is greater than even your marriage bond is down here. As to the ends of marriage, the view of Dietrich von Hildebrand is everywhere the standard. Sticking strictly to the question this is said to allow other marriages. There is no dogmatic de fide statement about what the relationship of former spouses is in heaven. I do personally like St Thomas More's
Here lies Joanna, dear little wife of Thomas More, who intends this tomb for Alice and me. The first united to me in my youthful days, gave me a boy and three girls to call me father. The second, a rare distinction in a stepmother, was as affectionate as if the children were her own. It is hard to say if the first lived with me more beloved than the second does now. Oh how blessed if fate and religion had permitted all three of us to live together. I pray the tomb and heaven may unite us, thus death could give what life could not give.
Epitaph written by Sir Thomas More for his wives seen to this day in Chelsea Old Church
Jesus did not exactly say marriage ends with death. He said:
For when they rise from the dead, they neither marry nor are given in marriage, but are like angels in heaven. -- Mark 12:25
Literally this means that a person cannot marry in the afterlife. It does not mean that a marriage is dissolved when a person dies.
Writing in the Catholic Standard, Msgr. Charles Pope writes:
... We ought not to conclude that a long marriage in this world will have no meaning at all in the age to come. In heaven our body and our soul will be perfected. So, too, our fundamental relationships, both spousal and familial. They will not be discarded or simply forgotten. Surely those who are spouses here will experience a far more perfect union in Heaven... While the juridical aspects of marriage may end at death, the union of hearts and lives will not.
The Catechism states in CCC ¶1026: “Heaven is the blessed community of all who are perfectly incorporated into Christ.” It is hard to think of a truly blessed community that does not include families. Thus Msgr. Pope's point is well taken.
We also have testimonies from saints who, shortly before death, reported seeing their families in the next world. For example, St. Thérèse of Lisieux promised her her sisters that she would "be even more with you than I was before. I’ll will not leave you. I will watch over Uncle and Aunt, over my little Lèonie, over all of you. When they are ready to enter heaven, I’ll go very quickly to meet them." Is there a reason why married saints (there are about 500 of them) would not watch over their spouses as well, and welcome them eventually into heaven if/when they arrive there?
However, for Catholics, the focus in heaven is not humans, but on God. People who reside in heaven are "Absorbed in the beatific vision," but not so much that they do not recognize their relatives and still have special feelings for them.
As Sr. Marie Morgan points out, we also have countless testimonies from Catholic laypeople who report visitations from family members, including spouses, who have passed away.
(There are) innumerable stories told by people even today who lie on their death beds, sharing with us their experience of dying. How many hospice workers can attest to the fact that many people claim to have seen or even spoken with their loved ones as they lay dying! For some, it is easy to “write off” these experiences as drug-induced fantasies or as hallucinations caused by the brain “shutting down.” However, when these experiences bring great peace and joy to agitated souls, I believe they point to the presence of some sort of heavenly intervention. As my own mother lay dying nearly two years ago, she smiled and exclaimed that she could see her brother “up there!” I have no doubt that she did.
And if a nun will know her mother in heaven, it must surely be that that a happily married wife will know her husband, and the husband will know his wife. Jesus did not not tell us that marriages end in heaven, he said only that we will not get married there. Those already married remain so, at least in heart if not in law.