If you do a bunch of a proofs/read a bunch of proofs, you get used to some common techniques, such as proof by contradiction, combinatorial proof (where appropriate), how to set up variables, which additional lines to add in a geometry proof, etc.

Here, rewrite the rational number as p/q with p and q both integers, q nonzero. If the irrational number is z then their sum is

x = p/q + z

Suppose x is rational, rewrite as a/b like above

a/b = p/q + z

Manipulate

a/b - p/q = z

(aq - pb)/bq = z

We have just shown that z is rational. This is a contradiction. QED.