In practice, most angels write checks into companies they hear about through their own network — a founder they used to work with, a deal a friend is already in, or a syndicate they follow. Deal flow is the whole game: the best investors see hundreds of opportunities a year and say no to nearly all of them. A typical angel might make 5 to 10 investments a year at $10K–$25K each, building a portfolio of 20 or 30 companies over time.
Angel investing is a power-law business, not an average-returns business. Most of your companies will return nothing, a handful will return your money, and one or two — if you're lucky and patient — return enough to make the whole portfolio work. That's why diversification matters so much: a concentrated portfolio of three startups is closer to gambling than investing. In the US, you generally need to be an accredited investor (roughly $200K in annual income or $1M in net worth excluding your home) to participate in most deals.