It would probably help to write the definition symbolically. If f is any function, let L(f) mean that exists. I'll write ~L(f) to mean the negation of L(f), i.e., that does not exist. The definition says that f is unhelpful if
∀g. (~L(g) => ~L(f + g)).
Note that the definition quantifies over all functions g, so the formula above is a property of f only. Thus, when we say that f is unhelpful, there is no g to talk about. This is analogous to saying that x is the least element of a set S if ∀y ∈ S. x <= y. Again there is quantification over y, so this is a property of x only. When we say that x is the least element of S, we don't have any y in the context of our discourse.
The theorem says that for any function f, if f is unhelpful, then L(f). One way to prove it it to use the facts that L(f) implies L(-f) for all f and that L(0). These are the only fact about L that are needed, the rest of the proof is simply a logical manipulation.