Social proof can get tricky trying to define it, but what you should know about it is that it is a social phenomenon dependent on the notion of normative social influence. That definition is still too wordy and complicated so let us simplify it with a scenario.
Let us say that you are visiting a country. The barrage of new sights and sounds might overwhelm you that you might get disoriented the moment you step foot in that new place.
So, how are you going to fit in? At this point, your brain will look for people to emulate behavior from. By observing how people behave in that situation, your brain can formulate decisions on how best to approach new surroundings and ideas.
In essence, normative social influence is about surviving by doing what everybody else is doing. Social proof in marketing works similarly.
If a person is making a decision whether or not to purchase something, they always look for references. What do others recommend? Is the person recommending that product credible? Can you see them using the product and is it effective? How a person answers these questions is but a simple manifestation of social proof.