You'll notice in the second picture (the OP posted) the man does not have his arms folded, and I'll restate the comment that other forms of prayer (clasped hands, or clasping hands with others in a circle) are not discouraged as long as the person is reverent. (see other LDS pictures) 1
Why do LDS cross their arms during prayer?
They are taught in primary how to get ready to pray by folding your arms, bowing your head, and closing your eyes.
There is no official reasoning that I can see other than that with folded arms children can not be as disruptive/un-revenant during prayer. When missionaries teach how to pray there is no mention of what the hands/arms do, leading me to the assumption that it doesn't matter as much as the prayer itself. 2
What is the origin of this practice?
There are legends that the pretzel was invented by European monks in 610 AD as a reward to children who learn their prayers. He calls the strips of baked dough, folded to resemble arms crossing the chest, 'pretiola' ("little reward[s]")... Moreover, the three holes within the pretzel represent the three persons of the Holy Trinity: God the Father, God the Son, and God the Holy Spirit. 3
Also, see this Q&A.
Why did the LDS church choose to deviate from the practice of folding hands and what was the purpose behind this change?
See first statement and first answer.
1 https://www.lds.org/media-library/images/christ-praying-nephites-henninger-art-37745?category=category/book-of-mormon-all-gospel-art&&lang=eng, https://www.lds.org/media-library/images/family-kneeling-prayer-889739?lang=eng, https://www.lds.org/media-library/images/quote-andersen-prayer-1173305?lang=eng
3 https://en.wikipedia.org/wiki/Pretzel#History note there are no primary sources for this legend