Could someone please explain the proof of Hotelling's lemma? According to https://en.wikipedia.org/wiki/Hotelling%27s_lemma, 

the maximum of the firm's profit at some output is given by the minimum of the difference between the profit and the revenue.

However, it does not sound right, as the difference of difference of the profit and the revenue is the negative of the cost of the input. The minimum would correspond to the maximum cost for input incurred. How would the maximum of the fir's profit at this point?

Also what does it mean by "at some output y*(p)"? Is y*(p) fixed here?
Well done for spotting this one. It is (was) simply a wrong statement. It is indeed the maximum and not the minimum of the expression as (was in the past) stated in the wikipedia article. 

Then the maximum of this expression is the maximum of $-C$ (Cost) which is equivalent to the minimum of $C$, and the equivalence with maximum profit holds under duality.