And she put the skins of the kids of the goats on his hands and on the smooth part of his neck.
This verse appears within the story of Jacob's tricking of the elderly, blind Isaac into giving him the patriarchal blessing instead of bestowing it on his older-by-mere-minutes twin brother Esau, who was the rightful heir. The "she" mentioned here is Rebekah, Isaac's wife and the mother of the two young men.
Part of the background of the story is that the two parents played favorites (Genesis 25:28): Isaac preferred Esau and his "manly pursuits," while Rebekah favored Jacob, who is described as "a mild man, dwelling in tents" (Genesis 25:27), suggesting that he was more refined and that his aptitudes were more mental than physical. This favoritism put the couple at odds on at least one score, who would inherit the patriarchy after Isaac's death. Isaac evidently thought Esau the better candidate, since he was the older and stronger. His wife felt Jacob better suited to the position, being more cunning and skillful in business and management. It also spurred rivalry between the sons.
Jacob had revealed his cunning when he had bargained the birthright from Esau some time before (Genesis 25:29-34). He made cynical use of Esau's famished state to finagle the lucrative--even precious--birthright from his brother, whom the Bible says did not value it highly enough: "Esau despised his birthright" (Genesis 25:34; see Hebrews 12:16-17). The birthright was the firstborn's double portion of inheritance. (Jacob later passed this birthright on to Joseph's sons, Ephraim and Manasseh; see Genesis 48.)
In Genesis 27, Isaac had sent Esau out to hunt for game to make his favorite stew, after which he would pronounce the blessing on him. Rebekah knew that this gave her time to make her own stew from the meat of young goats to imitate Esau's dish and to prepare Jacob to disguise himself as his hairy brother (see Genesis 25:25). Jacob was a "smooth-skinned man" (Genesis 27:11) by comparison to Esau, so he would need, not only to wear his brother's clothes so that he smelled like him, but also to apply hair to the backs of his hands and neck to make the ruse work.
So, Rebekah evidently adhered the skins of the freshly killed kids to Jacob's hands and neck, perhaps even sewing them to her son's cuffs and collar so that Isaac would never think that the hair he felt was not genuine. With the short time she had to work with, she went to great lengths to ensure that Jacob received the blessing--and even then Isaac nearly guessed the truth when Jacob could not imitate Esau's voice well enough (Genesis 27:22).
Perhaps what God said in Genesis 25:23 motivated Rebekah: "Two nations are in your womb, two peoples shall be separated from your body; one people shall be stronger than the other, and the older shall serve the younger." She knew that God had ordained Jacob to lead the family (see Malachi 1:2-3; Romans 9:10-13). However, like Sarah before her, Rebekah took matters into her own hands rather than allowing God to work matters out so that Jacob would receive the blessing in a more ethical way.
Who knows how He would have worked it out--maybe Esau would have despised the blessing too or Isaac would have been warned by God not to bless Esau but to bestow it on Jacob at a later time. It is a moot point now. God included it in His Book so that we can learn lessons from what actually happened--lessons about the use of trickery, favoritism in families, getting ahead of God, making assumptions about what He is doing, priorities, selfish ambition, parental manipulation of their children, how one lie begets another, and so forth. We can mine a wealth of wisdom from the rivalry of Jacob and Esau.