The short answer is because:-
It's fascinating and you might help to change the world (quite apart from possibly scooping the jackpot).
When we're in the twilight of our career or perhaps we've retired, some of us don't want to just stop working or feeling we're still useful. Whilst I love relaxing or playing golf in the sun, I don't envy the legion of prune-skinned Algarve émigrés. Through angel investing alone, I'm closely involved with VR, sports management, telecom surveillance, travel clubs, social media, parental control of kids' screen time, recruitment, health food, remote horse monitoring and many more weird and wonderful businesses. Beats agonising over whether a penalty stroke should have been added to someone's score card.
And it doesn't have to cost a fortune, especially if you get government incentivised tax relief. A typical angel invests £10-20,000 per business. Wealthier souls might consider £50-100,000 for larger amounts of equity (and income tax relief). (Note to my wife... It's called 'investing', not 'spending'). It's encouraged by HMRC because it's really important for as many small businesses to start up as possible, and banks won't lend without caste iron security. They also don't provide ongoing advice and support. So where else will people find funding to start a business, especially where there may be little evidence to prove it's low risk? And if there are already plenty of competitors, a small share of a big cake may still be worth going for, but traditional lenders don't like risks of any sort.
For me it all started by first becoming a business mentor. By the time I reached my early-40s I was already a serial entrepreneur. I had started trying to build my own businesses in evenings and weekends soon after I left uni, and while I was gaining experience at someone else's expense. So when I finally gave up full-time employment in my late 20s - or more accurately it gave me up, I had not only learned quite a lot about what it was going to take to do my own thing, I also knew something about how big business worked. But despite being incredibly busy building my own companies, I knew I could also find time to pass on some of the lessons I had learned.
I started really small by volunteering to mentor for The Prince's Trust who loan small amounts of money (although large by their clients' standards) for low risk business ideas like decorating, window-cleaning or vending street food. The money the charity lends goes to disadvantaged young people with a passion to stand on their own two feet (usually two), but who need their hands held until they become established. It's extremely fulfilling (although it can also be frustrating if your client, or their business, falls off the rails - which all comes with the territory). Mentoring's not hard for people who broadly know how business works:- Sales and Marketing, especially website creation, Google management, Finance (I usually recommended using part-time bookkeepers who knew more about accounting and especially tax than me), and other basic business functions. In fact you tend to learn on the job alongside your client about stuff like shop leases, casual employment law, grants etc. Where you don't know something, The Prince's Trust have someone who will.
After several years and a wide variety of clients, not all successfully launched as entrepreneurs, but all wiser for the attempt, I felt confident enough to mentor larger business - and often ones who needed investment to grow (or just survive, but disguised as growth potential). All of this I did whilst growing, and eventually selling some of my own businesses. Whereupon I also acquired personal experience in what's known as the 'exit'... Payday! Or, as it turned out, fuel for more start-ups (some of them testing my own ideas such as TrackBack and my latest experiment, TABi).
It never ceased to amaze me how much I learned from the businesses I mentored. It's easy to think that just because you've managed to found and grow a business yourself, you know the recipe for the magic sauce that made it happen. In fact, all you know, or think you know, is what helped to make your own business grow, not universally what makes all businesses grow. The principles might be similar - like don't try to sell what you want to sell, sell what people want to buy - but how they are applied will be infinitely variable.
Angels tend to hear about investment opportunities through clubs of like-minded people. I'm a member of The S100 Club based at the University of Surrey and the only angel network attached to the world's largest incubator network, SETsquared. We attract pitches from a wide variety of entrepreneurs trying to make a go of it in all sectors, but especially technology (the definition for which, in my opinion, is 'something that doesn't work properly yet'). By evaluating businesses as a group, we expose weaknesses faster, as well as identifying opportunities that perhaps the founders themselves haven't seen. We also represent a wide network of connections to short-cut introductions. And when we invest, we do so as a pack, so it's much cheaper and safer. The club itself also provides a certain amount of Due Diligence research about the companies, so it's not as Wild West as it might be if we were investing on our own.
Once the investment round is closed, the directors (usually) stay in touch with their investors by issuing progress reports, or sometimes cries for help. Some investors like to get more deeply involved, especially if they've invested larger amounts for larger slices of equity. They might become mentors to the chief executives or perhaps Non-Executive Directors (NED) and sit on the boards to watch out for investors' interests more intensely whilst offering varying amounts of hands-on support.
Angeling is a way to stay useful. To stay excited by the cut and thrust of business - whilst knowing that it's not our own arses on the line (or at least not much of our arses). To work very closely with amazing, passionate, bright young people. To become part of a solution that might benefit the world, but could at least provide employment and pay taxes. And maybe, just maybe, make enough money to buy a big villa in the Algarve on a prestigious golf course - for when you feel like getting bored.