Sorry, it's supposed to be the innermost Y+1 being the same, not Y-X+1 (which was the case when I was thinking about it as 1dX + 1dY instead). The outermost X-1 values on each side will have the standard sort of linearly-decaying distribution you normally see from rolling two dice.

As for why, consider, one die alone (specifically, the larger one). All values have an equal chance of coming up, from 1 to X+Y. Then add in another die, with equal chances of 1 through X. The larger die has Y more numbers than the smaller die that it can generate. The innermost combinations of values (in a simple example of 1d4+1d10, those would be 5 through 11) can be generated using any values on the smaller die and an appropriate value on the larger die, but the outermost values (2, 3, 4, 12, 13, and 14) can only be generated using some values on the smaller die; others do not have an appropriate value on the larger die with which to generate them. For example, 7 can be made with 1+6, 2+5, 3+4, and 4+3, by 12 can only be made with 2+10, 3+9, or 4+8; a 1 on the d4 doesn't do it. So, the innermost values all have the same number of chances of coming up (4/40, in the example above), but the outermost values have the usual linearly decaying probabilities that you would expect from rolling two dice together.

Edit: A bit more math fail. It's X-1 on each side, not X in total.