Can anyone explain this more clearly?

Question

I'm new to CGT so i might need help but could anyone simplify this and explain it to me please- "set f ⊕ f = 0 for any f. (A nice correspondence can be made if we think back to the original game of Nim: if you take any two equal size Nim piles and add their values via binary addition without carry, you always obtain zero!) From this, it immediately follows that F, ⊕ constitute an abelian group where every element is of order 2. If F were finite, then the remainder of our argument would follow directly from the fundamental theorem of finitely generated abelian groups, which states that G must then be a direct sum of additive groups of integers modulo k; however, since this group must be of order 2, all the k’s must be 2 (since if there were a k > 2 then there would be an element of that order, and not of order 2.) When F is infinite, then the conclusion is still guaranteed since any group of bounded order (i.e., where there is a bound on the orders of all group elements) is a direct sum of cyclic groups [10]. Recall that the additive group of integers mod 2 is simply 1-bit binary addition without carry, so the above characterization can be interpreted as F being the binary sum mod 2 without carry of integers. Thus, we get the basic structure of the Sprague-Grundy theory from a decomposition theorem about groups and the mystery of the base 2 is revealed." I dont properly understand the fundamental theory of finitely generated abelian groups so if you could simplify that that would be helpful.