A link's already been posted to this recent answer by /u/toldinstone, which talks about some specific philosophers. I just wanted to shift the focus a little bit and talk about what made it possible for these people to spend their life thinking and asking questions.

In most modern societies, our concept of the good life is tied up with industry: the harder you work, the more you deserve wealth and status. In practice it doesn't really work that way, but at some level we all believe it. We think that if we want a better life for ourselves, we need to get off the couch and work for it.

The ancient Greeks did not draw the same link between working and living the good life. In their view, the ultimate goal of the virtuous citizen was to get to a point where he would not have to work for a living. Greek societies were divided into "the rich," who did not work, and "the poor," who had to work. In many oligarchic states only the leisure class had political rights. Sparta was an extreme example in that citizens were actually banned from having any profession. Philosophers like Plato and Aristotle argued that anyone who had to spend time working with his own two hands was not suited for citizenship. After all, he would not be able to devote all his time and energy to the government and defence of his state.

This attitude to wealth and good living creates a very different dynamic than the one we're used to. Our image of a rich person is more or less constantly involved in the business of making money, by whatever means they do so. By contrast, Greek society contained a fairly broad group of people who had already attained the level of wealth they were aiming for, and now lived a life of leisure - explicitly detached from whatever was sustaining their lifestyle. Each of these people filled their time as they pleased. Some would focus on personal pleasures like hunting, horse-raising, patronage of the art, or patronage of sex workers; others might be interested in serving the public as politicians, orators, generals and soldiers. Philosophy was another one of the available options. You could choose to spend your time thinking and talking to other thinkers and figuring out higher truths. If your status was not so secure, you could make extra money selling your knowledge to others.

This entire lifestyle was predicated on the idea that someone else, somewhere else, was working on your behalf. Philosophers made their living in the same way as the rest of the Greek leisure class: by exploiting the labour of others.

The traditional form of wealth in Greece was land. This land was worked by poor tenants and enslaved labourers. Sources like Hesiod's Works and Days and Xenophon's Oikonomikos make clear that the wealthy estate owner would not be doing the work with his own hands, but would buy himself an enslaved overseer to manage the work of his enslaved workers on his land. In other words, his ability to live the good life was predicated on the fact that 100% of the productivity of many other people flowed directly into his storage rooms.

By the Classical period there were many other ways to sustain a life of leisure. Some took out mining contracts; others invested in shipping or loaned out their cash against interest. Many people were kept in the leisure class by the maintenance of large groups of enslaved labourers, either working together in a "factory" (such as the shield factory of the father of the orator Lysias, or the furniture factory of the orator Demosthenes), or simply rented out to whoever needed additional labour. These people produced a surplus they would never benefit from; instead it allowed their enslaver to indulge in his pleasures and possibly spend his days walking around the agora or the gymnasion talking to folk like Sokrates.

In short, our question should not be "what did these philosophers contribute to society that allowed them to make a living?" It should be "Whose labour was making these philosophers' living for them?" The answer, as usual in ancient Greece, is vast numbers of enslaved human beings for whom Plato's cave was anything but an allegory.

Here is a great answer to this question thanks to /u/toldinstone.