The 19th century saw the professionalization of science in a major way. What this means is, the number of people doing science increased, the amount of funding for science increased, and there became a "career path" for science that didn't involve being extraordinarily lucky or extraordinarily wealthy. Accompanying all of this was a radical change in scientific institutions: in this period, one sees the rise of industry-supported science, state-supported science, and the emergence of the German-model research university where research was subsidized with student tuitions.

It also helped that the groundwork for several new, very productive means of research were laid: the Chemical Revolution had created a whole new path for thinking about what was going on in the elements; developments in new instruments and methods lead to huge revolutions in physics and astronomy; and the emergence of reliable global communication allowed for the kinds of long-distance correspondence networks necessary to sustain truly large-scale discussions about biology, geology, paleontology, and so on. All of this means that new opportunities opened up and once they did there was a lot of "low-hanging fruit" of new ideas, discoveries, etc., to see.

So the answer to your question is neither just people being motivated (there was also the birth of "Scientism," which is to say, the near-religious belief that "science held all the answers," and that it was a noble vocation, etc.), or money (though money is the fluid that greases the wheels in most cultures, so these new ways of routing money towards science certainly played a necessary role), but rather the emergence of what people today consider "science" to even be: not just a few tinkerers or geniuses or wealthy obsessives, but a large community of researchers with access to resources tackling many problems in a systematic way. Though this came of age in Western Europe, it would, both because of Western Europe's global dominance but also because the this approach gets much more productive results than any competitors, be exported globally, to a degree that it no longer makes any sense to think of this as a purely European model.

What surprises people about this explanation is that most people are told that science emerges with the so-called Scientific Revolution of the 16th-18th centuries, and so the 19th century seems rather late for it to really come onto the scene. But this is the difference in thinking about science not as a methodology (it has no single methodology), or a collection of geniuses (Copernicus, Galileo, Newton, oh my!), but rather as a social enterprise. At best you can say that the Scientific Revolution laid the intellectual, social, cultural, and institutional foundation for what would only really be realized in the 19th century: the emergence of science as an profession. It is not a coincidence that it was only in 1834 that a new term was suggested for the people who took in that profession: "scientist."

The other surprising fact: for all of the successes and support of 19th century science, it was still, by the standards of the 20th century, a small and anemic enterprise. The 20th century would see all of these trends in science expand by orders of magnitude. But the systems that we have today are basically recognizable in the 19th century, whereas they are only barely recognizable earlier than that.