.Let G be a finite group and H a subgroup of G. Let N(H) = {a∈G: axaˉ¹∈ H for every x∈H}. Prove a∈N(H) implies that aH aˉ¹∈H.

This is the exact definition of you wrote three words before. There's nothing to prove here.

Prove also that N(H) is a subgroup of G. Prove that H⊆N(H); indeed, H is normal subgroup of N(H)

This is just applying both the definition of normalizer and of normal subgroup. If you have doubts re-read the definitions and understand them.

Tonio