What this Southern Baptist understands from the passage is several things, and it is necessary to include the entire exchange to understand it. Let's take the verses you referenced and see what Jesus was conveying to his disciples.
Matthew 19:1 And it came to pass, that when Jesus had finished these
sayings, he departed from Galilee, and came into the coasts of Judaea beyond Jordan;
In chapter 18 Jesus had just given them a lecture on how they were to treat one another, and the essence of forgiveness.
Matthew 19:2 And great multitudes followed him; and he healed them there.
Jesus had further emphasized his message by healing a large number of people, He did not question each of them to determine whether or not to heal them he did it purely out of compassion for the people.
Matthew 19:3 The Pharisees also came unto him, tempting him, and saying unto him, Is it lawful for a man to put away his wife for every cause?
Then the Pharisees who were adversarial to Jesus, who was teaching things contrary to the rigid procedures they were putting on the people.
Matthew 19:4 through 6 And he answered and said unto them, Have ye not read, that he which made them at the beginning made them male and female, 5 And said, For this cause shall a man leave father and mother, and shall cleave to his wife: and they twain shall be one flesh? 6 Wherefore they are no more twain, but one flesh. What therefore God hath joined together, let not man put asunder.
Jesus answered them in the same manner he resisted Satan, by using God's own words to counter their attack.
Matthew 19:7 They say unto him, Why did Moses then command to give a writing of divorcement, and to put her away?
Moses was revered among the Jews, and the Pharisees apparently thought if they could get Jesus to contradict Moses they could cause the people to reject Jesus.
Matthew 19:8 and 9 He saith unto them, Moses because of the hardness of your hearts suffered you to put away your wives: but from the beginning it was not so. 9 And I say unto you, Whosoever shall put away his wife, except it be for fornication, and shall marry another, committeth adultery: and whoso marrieth her which is put away doth commit adultery.
Here is the key to understanding Jesus lesson, in that He is explaining that marriage is a Holy union sanctioned by God, and just as God only made one woman to be Adam's helpmate, God intended that once married; then the two of them were inseparable.
Matthew 19:10 His disciples say unto him, If the case of the man be so with his wife, it is not good to marry.
Verse 10 shows us that even his own disciples were confused by that saying, even to the point that they were thinking if you can never be parted from a woman after marriage it is far better to just not get married in the first place.
At this point it is necessary to say that verses 8 and 9 are read very differently by even people in the same Denomination; Some believe that he is saying adultery is the only reason for divorce; while others believe that it is saying if the man divorces a woman and she remarries he has caused the woman to commit adultery.
Matthew 19:11 But he said unto them, All men cannot receive this saying, save they to whom it is given.
Jesus is here telling them that this will only make sense to those to whom God has determined that they should remain unmarried and that is done in three ways.
Mat 19:12 For there are some eunuchs, which were so born from their mother's womb: and there are some eunuchs, which were made eunuchs of men: and there be eunuchs, which have made themselves eunuchs for the kingdom of heaven's sake. * He that is able to receive it, let him receive it.*
which were so born from their mother's womb: This is a hard one to understand, but basically what Jesus is saying is that some men will be born with physical or mental deficiencies which preclude their being able to marry.
which were made eunuchs of men: notice that he uses the past sense verb form "were made" In the old testament men of captured nations were castrated for several reasons one very common one was as Harem guards.
which have made themselves eunuchs for the kingdom of heaven's sake. Here he uses the present sense of the verb form, the significance of this is in that it is a voluntary thing, which the man chooses for himself. That can and sometimes does change. There are instances where Priests have left the priesthood and married.
The last part * He that is able to receive it, let him receive it.* is his way of saying this life is not for everyone, but those who can control their sexual urges to live a single life let them make their own choice.