Mathematician David Henderson explains that: A proof
is a communication -- when we prove something we are not done until we can communicate it to others and the nature of this communication, of course, depends on the community to which one is communicating and is thus in part a social phenomenon.
is convincing -- a proof "works" when it convinces others. Of course some people become convinced too easily so we are more confident in the proof if it convinces someone who was originally a skeptic. Also, a proof that convinces me may not convince you or my students.
answers -- Why? -- The proof should explain something that the hearer of the proof wants to have explained. I think most people in mathematics have had the experience of logically following a proof step by step but are still dissatisfied because it did not answer questions of the sort: "Why is it true?" "Where did it come from?" "How did you see it?" "What does it mean?"