Background for Hejhal's "The Selberg Trace Formula for $PSL(2, \mathbb{R})$

I'm trying to learn the Selberg trace formula, but have very little background. I was referred to Dennis Hejhal's The Selberg Trace Formula for $PSL(2, \mathbb{R})$ but just got the book and was dismayed to learn that that the author assumes familiarity with Selberg's original paper (which I don't have access to).

There's a bunch of stuff in the first few pages that I don't know. For example, he states without proof that the spectrum of the Laplacian on a compact hyperbolic surface is discrete, and one of the things that I would like to understand is why this is so. He gives a reference to a 1912 book by Hilbert, but aside from the fact that I don't read German, it's not at all clear to me that this is the best place to learn from (in light of the fact that Hejhal's book is from 1976 and many books have been written since).

Does anyone have a suggestion for what to read before Hejhal's book?

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