Why only in the last few hundred year has science and technology progressed like it has

Larger population means that even if the percentage of people with the necessary skills and interests is the same, the total number of people with those skills and interests is much greater.
Combined with the ever increasing base of scientific data on which to build (as pointed out by Andy) this causes an increase in the potential for scientific progress (and progress in other fields like engineering to go along with it).
Of course if you have a society that actively discourages interest in science and engineering that will cause a slowing of that progress, and we seem to be entering such an age. But that happens occasionally throughout history, and there's still the mechanism of greater numbers in place to make progress faster relative to those periods.

One thing that could (and at times has) caused things to seemingly slow down is the requirements for ever larger teams and investments needed to make new discoveries. It was quite possible for Christian Huygens to design, fund, and create his first microscopes. For the person coming up with the idea for the electron microscope that was already a lot harder. And to build a cyclotron on your own is just about impossible unless you happen to be a rather affluent person with advanced skills in many fields.
But overall, it holds true that more people can get more done in total in amount of time than a smaller group.

There's just more scientists around today. Population has grown tremendously over the past 200 years.

On top of that, our means, tools, and time available to spend on research has grown too. And much faster communication allows ideas to spread much faster, allowing other scientists to build on what you did immediately, rather than 10 years later.

At the same time, don't underestimate the progress of the past. The middle ages are often seen as a time when not much progress was made, but there were tremendous advances in construction, defense works, metallurgy, weapon design, etc. Not to mention the introduction of a totally new number system, sailing, compass, etc.

New technologies build on previous technologies, so technological growth is cumulative.

The rate at which it is developed, and hence accumulated, depends upon the number of people who can work on producing new technologies. The number of people engaged depends on technologies that give a proportion of the population the time to do things that are not directly related to survival. As more technologies are produced, a greater population can be supported with less effort per person.

A specific example of how technological advancement depends on earlier technologies is how writing changed things. In prehistorical times advancement relied on accident and insight. But with only word-of-mouth and direct teaching, progress was slow and knowledge could get lost. After writing, discoveries could be recorded and disseminated.

A narrative argument is

• spears allowed easier and safer hunting, increasing the population
• some of the increased population could spend some time developing new technologies (over millenia)
• one of the new technologies was domesticating food production animals, supporting an increased population with even less effort per person. Roles could become more specialized, and people who specialize can focus on solving specialized problems.
• the new technology supported larger population centers (towns and cities) and larger ruling classes, some of whom had more leisure to spend on developing new technologies. The ruling classes also had enough resources to employ specialists, such as military, and to undertake larger projects, such fortifications and irrigation.
• about 400 years ago, a tipping point was reached, and the seeds of the industrial revolution were sown. The industrial revolution when it arrived freed huge fractions of the population to specialize even more.
• 200 years ago the current technological snowball effect you describe took off, based on the previous million years of progress.