Methods for proving upper bound on a-approximiation algorithms?
You mention two results giving tight bounds on the approximability of MAX-E3SAT: an upper bound (algorithm) and a lower bound (hardness). The upper bound is much simpler than the lower bound in this case. Indeed, using basic probability you can show that a random assignment satisfies $7/8$ of the clauses in expectation, which gives an approximation algorithm whose approximation ratio is at least $7/8$ in expectation. If you don't like randomized algorithms, this algorithm can be derandomized using the method of conditional expectations.

The other direction is much harder, and relies on the PCP theorem and Raz's parallel repetition theorem. It was shown by Håstad in his seminal paper Some optimal inapproximability results. This is all described in Arora and Barak's textbook, but I warn you that the textbook is quite advanced.