His first proof uses mathematical induction. I started to compose a reply in which I explained mathematical induction, but I can't help thinking someone, perhaps AlexB on this very website, has explained it better than I can in a hurried reply. Math induction is so useful that you should have it explained carefully. At first glance, especially if it is not explained well, it looks like circular reasoning.

Anyway, the induction hypothesis in the first proof was that
2n-4 was an upper bound for the number of calls needed to fully share information among n people. Because it was the induction hypothesis, he could use it to show that 2(n+1)-4 is an upper bound for the case of n+1 people.

Although he alluded to the crucial first step of his induction proof, he did not go into the details. The first step (which also answers your second question) is to prove that 2k-4 is an upper bound for some number of people, in this case, 4.

Using Manoj Nair's notation:

12, 12, 3, 4
12, 12, 34, 34
1234, 12, 1234, 34
1234, 1234, 1234, 1234

Four people, four phone calls.