Rationality is essentially human logic plus empirical facts. Science is essentially a systematic and rigorous application of human rationality. The scientific method is therefore rational. Thus, rationality precedes science but science is designed to perform better.
There are several important empirical facts concerning rationality. First, it is possible to distinguish between rationality and irrationality. Second, we can often assess whether our actions produced a bad result or a good result. Third, rationality seems to work better than irrationality. Fourth, where it is used, science often can be seen to work better than ordinary rationality.
Thus, rational people accept that science is often the best way to be rational. This is not always true, though. Ordinary rationality is good enough for our everyday life. To use the scientific method in every circumstance of your life would be silly because in many cases science wouldn't give results fast enough or even wouldn't be able to give any result.
And this is all there is to it. Use what works best.
We may wonder that rationality should work at all. Well, when we were ignorant of the science of that, the puzzle was legitimate. Nowadays, we understand why rationality works. Essentially, rationality is a natural capacity we inherited through our DNA, capacity produced by natural selection. Natural selection eliminates characteristics that don't work well enough. We have the logic we have because it is the logic that was good enough to allow our ancestors to survive in their natural environment. If in the past the logic of some people was not good enough, they found it difficult to survive and reproduce. They didn't pass on their logic to us precisely because their logic was not good enough. Thus, it is no surprise that rationality works.
Thus, since rationality works, it is no surprise that science should work, and should work better since it is only rationality on steroids: more rigorous, more collaborative, more organised, more systematic, less forgetful, and often given more resources.
This is all a rational mind needs to feel confident about science.
Rationality does not require that we should be able to explain the whole of reality to accept that science works best, at least in some very specific domains. Rationality only requires that we should use the means best suited to our objectives. Science does not pretend to explain reality. Instead, it is our best method to predict the outcome of real processes, and therefore predict future events. Thus, we should use science whenever we need a method to predict such events.