So the latter day saints are saying that God planned for Adam and Eve to fall from grace and bringing sin into the world?
Yes, basically. The Book of Mormon, 2 Nephi chapter 2, explains this. The whole chapter is fantastic, and quite the sermon of one of the prophets to his sons, so I recommend reading the whole thing because it's hard to pull these verses out of context. Still, this is the best I can do and I think gets at the crux of your question:
22 And now, behold, if Adam had not transgressed he would not have
fallen, but he would have remained in the garden of Eden. And all
things which were created must have remained in the same state in
which they were after they were created; and they must have remained
forever, and had no end.
23 And they would have had no children; wherefore they would have
remained in a state of innocence, having no joy, for they knew no
misery; doing no good, for they knew no sin.
24 But behold, all things have been done in the wisdom of him who
knoweth all things.
25 Adam fell that men might be; and men are, that they might have joy.
26 And the Messiah cometh in the fulness of time, that he may redeem
the children of men from the fall. And because that they are redeemed
from the fall they have become free forever, knowing good from evil;
to act for themselves and not to be acted upon, save it be by the
punishment of the law at the great and last day, according to the
commandments which God hath given.
I'm going to add Alma 42 as well, and do my best to choose the most relevant excerpt (although the whole chapter is important for this topic):
11 And now remember, my son, if it were not for the plan of
redemption, (laying it aside) as soon as they were dead their souls
were miserable, being cut off from the presence of the Lord.
12 And now, there was no means to reclaim men from this fallen state,
which man had brought upon himself because of his own disobedience;
13 Therefore, according to justice, the plan of redemption could not
be brought about, only on conditions of repentance of men in this
probationary state, yea, this preparatory state; for except it were
for these conditions, mercy could not take effect except it should
destroy the work of justice. Now the work of justice could not be
destroyed; if so, God would cease to be God.
14 And thus we see that all mankind were fallen, and they were in the
grasp of justice; yea, the justice of God, which consigned them
forever to be cut off from his presence.
15 And now, the plan of mercy could not be brought about except an
atonement should be made; therefore God himself atoneth for the sins
of the world, to bring about the plan of mercy, to appease the demands
of justice, that God might be a perfect, just God, and a merciful God
16 Now, repentance could not come unto men except there were a
punishment, which also was eternal as the life of the soul should be,
affixed opposite to the plan of happiness, which was as eternal also
as the life of the soul.
17 Now, how could a man repent except he should sin? How could he sin
if there was no law? How could there be a law save there was a
18 Now, there was a punishment affixed, and a just law given, which
brought remorse of conscience unto man.
19 Now, if there was no law given—if a man murdered he should
die—would he be afraid he would die if he should murder?
20 And also, if there was no law given against sin men would not be
afraid to sin.
21 And if there was no law given, if men sinned what could justice do,
or mercy either, for they would have no claim upon the creature?
22 But there is a law given, and a punishment affixed, and a
repentance granted; which repentance, mercy claimeth; otherwise,
justice claimeth the creature and executeth the law, and the law
inflicteth the punishment; if not so, the works of justice would be
destroyed, and God would cease to be God.
23 But God ceaseth not to be God, and mercy claimeth the penitent, and
mercy cometh because of the atonement; and the atonement bringeth to
pass the resurrection of the dead; and the resurrection of the dead
bringeth back men into the presence of God; and thus they are restored
into his presence, to be judged according to their works, according to
the law and justice.
24 For behold, justice exerciseth all his demands, and also mercy
claimeth all which is her own; and thus, none but the truly penitent
25 What, do ye suppose that mercy can rob justice? I say unto you,
Nay; not one whit. If so, God would cease to be God.
26 And thus God bringeth about his great and eternal purposes, which
were prepared from the foundation of the world. And thus cometh about
the salvation and the redemption of men, and also their destruction
Sorry for the wall of text. But you asked a question where the answer is the whole Plan of Salvation, which has a lot of substance to it. These passages are saying that we were consigned to fall out of God's presence, to be tried and tempted, and learn from our own experience, and then to be able to prepare to meet God and become like Him, whereby we'd be redeemed from the fall and from our own sins through the Atonement of Jesus Christ, and to be brought back into His presence by the power of the resurrection. Without the Fall, none of that could have happened.
According to this theology, Jesus Christ was not a backup plan, He was always The Plan.