For a really simplified example. Let's say there's a company that simply holds $100. No revenue or expenses. It's got 100 and that's it.
And say there are two shares total and two shareholders. Me and some other guy. So each share is worth $50.
Say the company does a buy back and spends 50 to buy the other guys share. Now I own the whole company but it's only worth $50. So I'm no better off than before.
What am I missing?
Nothing. Buybacks are a way to return cash to shareholders, just like dividends, with different tax implications.
In your example the company has used the share buyback to return 50$ to shareholders. If it wanted to do it evenly it could even do a stock split first turning every 50$ share into two 25$ shares and then buy back one share from each person. That would be functionally equivalent to doing a 25$ dividend per original share but have different tax implications (capital gains are cheaper than income taxes in most places).
You're just looking at yourself and not the entire class of shareholders. Buybacks are positive for shareholders who want to cash out. As well, the additional demand for shares tends to boost the share price.
In a contrived example where a company is totally valued by its cash balance, you have the appearance of a wash. But if a company's worth is determined by revenues exceeding expenses (profit margins) rather than cash balances, you can boost the share price beyond e.g. $50 with the increased demand, where the share price is not a simple function of cash balance but has expected cash flows as a major component.