Existence of periodic orbits in rational billiards

Schwartz, Richard Evan. Mostly surfaces. Vol. 60. American Mathematical Society, 2011.

On p.219ff of Schwartz's book, he sketches "an elementary proof, due to Boshernitsyn, that every rational polygon has at least one periodic billiard path."

(He doesn't cite an explicit reference for Boshernitsyn's proof.)

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In answer to "what background is needed...?": The first 218 pages of Schwartz's book. :-) Less flippantly, you could essentially read backwards from the proof sketch to the concepts that precede it.

You may go directly to the original result by Mazur himself

H. Masur, Closed trajectories for quadratic differentials with an application to billiards, Duke Math. J. 53 (1986), 307-313

or a stronger theorem on the density in this Transactions AMS paper.

For a good read and background take a look at the book by Hurt, "Quantum Chaos and Mesoscopic Systems: Mathematical Methods in the Quantum".