Why do companies want to issue shares in the first place? Isn't it better to just maintain private ownership?

First and foremost, an initial public offering (read: IPO) allows liquidity for early investors and shareholders of the company.

While it is true that founders usually see the benefits to maintaining private ownership (and not be privy to public scrutiny as compared to private companies) - they would still need to consider what the private shareholders want.

Secondly, share issuance is a means to raise capital quickly for the company.

As you can see with the recent IPOs, the amount raised is sizeable. Not only does the company amass funds, it also gains publicity which would be important for the business as well.

Prestige attached to listing on a major stock exchange

You probably might have seen those photos where a company lists on a major stock exchange and the smiling exec team is there to cut the ribbon/ring the bell etc. There is still the vanity value to listing on a major stock exchange.

I think Enk Loui has provided a rather comprehensive explanation as to why firms would want to issue shares, but I would also like to add on the reason why so many firms seek to do IPOS: it materializes their piece of the pie, and puts a quantifiable value to their company. While companies remain private, exit opportunities for these private equity shareholders are painfully little and it is very hard to cash out on their hard work, but when you offer publically to the exchange, you get to easily increase the liqudity of the shares, and since each share are traded on the open market with a particular monetary value attached to each share you get to easily see you net worth, holdings value etc. This is why when firms IPO, the leaders become millionaires!