So that the company can invest in future products and sell them to customers. If successful the investors make money, the company pays developers and customers enjoy whatever games the company makes.

The money is not 'public' in the sense of belonging to the government, but public in the sense that anyone can invest through the share market.

This is a much too deep a question to adequately address in this medium. A better starting question might be:

Do capital markets have a natural or a constructed social utility?

If their social utility is natural, then finding out what benefit they might have to society is a sort of empirical question. But if their utility is a purely political construction, then their perceived benefit can be moulded to fit a desired outcome.

Is it supposed to benefit society? The benefit is that their shareholders will make money - they're a corporation.

They don't just "receive" money they sell things for it. People vote with their money. That's capitalism.

IPO is an acronym for initial public offering. In this case, public refers to individual investors, mutual funds, etc. that have the opportunity to purchase equity in the company on stock exchanges like the NYSE.

No government funds are invested (unless you count public pension investments like CalPERS) during an IPO.

Actually, states like California stand to benefit from King's IPO by way of capital gains tax revenue.