So the question of patronage of science is perhaps the question for understanding historical (and, consequently, contemporaneous) scientific work. Since at least the 1930s, it has been how most historians of science even begin to approach understanding the scientific activities, motivations, and discoveries of the past, because it is just that fundamental. So it is a good question, but it also means that the answer is going to be different for nearly every scientific era and personality you could want to inquire about. It is also pretty core to the definition of what we call "science" in the first place, because it is at the heart of the institutional communities of people who seek to expand knowledge about the natural world.

So, for example, if you go back very far, to the early civilizations, thousands of years back, the pursuit of knowledge about the natural world (which I will use instead of "science" to avoid tricky issues about what is and isn't "science") was directly linked to their administration of bureaucratic, agricultural, hydrological civilizations. Their whole societies were arranged around these priorities, and natural knowledge acquisition was simply another part of state activities. So it is not too surprising to learn that the priorities of these investigations are linked to the practical problems that these states were trying to solve. Egyptian and Mesopotamian geometry, for example, emerged out of the need to measure land parcels, an issue of great importance when floods periodically wipe out boundaries between them. ("Geometry" means, literally, "earth measurement.") By contrast, in Ancient India, their geometrical investigations were rooted in different practical problems: the construction of brick buildings, which was a major activity in even pre-urbanized India. In both of these examples, what started as a very practical activity could also evolve into something more like an abstract activity — both cultures ended up developing mathematics that went beyond their immediate practical needs — but that is the root of the effort. Other "sciences" pursued by these sorts of cultures included things that we might not today recognize as "practical science," like astrology, but since the study of the heavens was believed at the time to actually foretell the future, they were quite practical indeed to those people. Hence the immense infrastructure (jumping forward quite a bit in time!) of, say, Song dynasty imperial astronomy in China, which had multiple observation hubs staffed with astronomers and instruments and mandarins (educated bureaucrats) who could do calculations, paid for by the imperial state, so that they could properly divine the will of heaven and thus give their leaders the "intelligence" they needed to make informed decisions.

Jim McClellan and Harold Dorn call this "mode" of science Babylonian as just a simple way to create a category that can encompass the similarities between these disparate states (Imperial China is a little different, but I digress). One interesting aspect of it is that in most cases we don't really know who the scientists were, because credit was not really an important part of the work. You weren't doing science for yourself, you were doing it for the society as a whole, and it was not a system in which you would become famous or wealthy by discovering something new. You could think of this as "just another government job."

OK, jump forward a bit. In the medieval period in Europe, who might be doing "science"? The answer varied a lot by place and time, but generally speaking the largest single "patron" of any kind of natural investigation would have been the Catholic Church, who was interested in the natural world both because it had some practical applications to their own work (e.g., accurate calendars are important to identifying feast days, and they often provided medical services to people, so herbal knowledge was important), but also because there were theological implications via "natural theology" — if you believe that God created the natural world purposefully, then studying the natural world is a way to study God. The side-effect of this is that where you do find people studying the natural world in this time period, they are often in monasteries, or associated with the Catholic orders that have most aligned themselves with natural theology, like the Jesuits.

Outside of the Church, the place in this world where one finds people doing "science" tends to be physicians, because what we would today label as chemistry, alchemy, and even astrology were often intermingled with medical practices. Such people might be independently wealthy, but they could also have established themselves as the physician to some wealthy or powerful person — sometimes as a courtier (part of a "court"). This kind of a patronage lasted well into the Early Modern period. Galileo famously named the moons of Jupiter after the Grand Duke of Tuscany, Cosimo II de' Medici, in an attempt to get a job as his court astronomer/mathematician. Galileo had been, until that time, working at the University of Pisa, but universities in medieval and early modern Europe were generally not sites of real research, but were primarily places of instruction. Galileo got the job, and with it he got the financing he needed to do independent research, buy materials for telescopes, publish pamphlets and books, and so on; he also gained some political independence in the process, because the Medicis were tough cookies (one of the reasons he had such a disastrous fall from grace many years later was because he was essentially patronless by that time, and that made him very vulnerable).

In this kind of model, interestingly enough, credit is very important — Galileo needs you to know that he, Galileo, discovered what he discovered, because his income and independence depends on him getting the credit for it! I bring this up just to highlight how the patronage system changes dramatically the "mode" of science that is being done, and systems that give individual "reward" for new discoveries end up changing the incentives structure quite a bit.

There are many other examples one could bring up. Tycho Brahe is an interesting case from the early modern period, because he did have a patron of sorts (King Frederik II of Denmark), but he was also independently wealthy. So he was able to do science on a very big scale for his time, basically having the equivalent of two astronomical laboratories on his own island (which the King gave him), with a relatively large staff to maintain them and help do astronomical observations, that would have otherwise been impossible for someone to do. But when he fell out of favor with the next King, he had to go into exile and search for a new patron (which he found in Holy Roman Emperor Rudolph II). So you can see there are some demerits to this system as well! (The number of stories of famed scientists who lost a patron towards the end of their lives and then were incredibly vulnerable is pretty large — Brahe, Galileo, Leibniz...)

One of my favorite things to do with students is to show them works that really emphasize the patronage issue. The Royal Society of London, for example, was created by King Charles II due to his interest in alchemy, and part of the royal charter meant that they could get some funds for printing their work (distribution of said knowledge is core to what we call "science"). And so it is not super surprising to see that the introduction to one of their most famous publications, Robert Hooke's Micrographia (1665) starts out with a literally oversized compliment to their patron, explaining how amazingly unfit the author of the book is to presenting his observations to so beneficent a patron. One finds similar things in vastly different time periods and locations as well regarding individual patrons. al-Khwārizmī's famous book on al-jabr (algebra), The Compendious Book on Calculation by Completion and Balancing (~830 CE) starts off with an introduction praising the Caliph, Iman al Mamun, who al-Khwārizmī claims encouraged him to write his volume. This is a much longer topic, but practical secular knowledge, especially translated from other cultures, was highly valued during the Abbasid Caliphate (the so called "Islamic Golden Age"), and wealthy figures and religious-political leaders sponsored large institutions (like the famous House of Wisdom in Baghdad) as well as individuals to compile, translate, and innovate on such knowledge. al-Khwārizmī's book is meant to be practical: all of the examples in it are about how to calculate taxes and interest and do trade (including some ones that are very reflective of the larger society, like how to deal with inheritance of slaves), but again, that is part of the larger superstructure that is rewarding this kind of work.

I've just given a few sketches to illustrate a few commonalities and differences, and to emphasize the hugeness of the question. It is core to the development of "science." Generally speaking, you did not have anything like independent scientists going to people for "funding" in the past. Either they were part of an existing bureaucratic apparatus, or they were able to do their work because they were independently wealthy or happened to work for someone with money and power. There were, at times, some institutions dedicated to furthering knowledge; the famous Museum of Alexandria (which included the Library), the Houses of Wisdom, the imperial observatories in China, etc. Though the work in these places was still molded by the reasons that someone (states, wealthy individuals, etc.) were financing them in the first place.