Why is Y a Convolution of X1 and X2 PDFs?

[reddvoid]

if X1 and X2 are two uniformly distributed random variables
and if Y = X1 + X2
why is that the probability density function of Y is convolution of probability density functions of X1 and X2 ?

I tried many ways, I'm not able to get at this conclusion

 

[D H]

http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter7.pdf

See Theorem 7.1, just after the start of section 7.2.