There are two ways to compensate with stock: purchasing stock on the open market with cash, or, hand out stock from a pool of withheld shares. The pool of shares might be periodically increased through creating new shares (and diluting) but this must be reported to the SEC and is only done infrequently.

GAAP reporting requires that stock compensation is written off as an expense. It does not reduce the company’s cash, but it does reduce the company’s earnings. Hence it changes the price to earnings figure.

Its from Accounting assets = liabilities + equity. So equity = a-l. If you pay cash your assets go down so your equity goes down. If you give away equity your equity goes down. In either case you equity goes down.