Point 1: increased demand for stocks will increase the price of it and thus the market cap. Point 2: the higher the market cap of a company the more capital it has available.
Yes, it theoretically means they could raise more money by issuing shares, but that's not happening.
But they don't have to do that. They could issue the shares for RSUs. If they issue new shares (either to sell to the market or to use for RSUs) then the market cap directly influences how much they can generate in work or cash from such a transaction.