They used the scientific method.

There is no one scientific method, so it's difficult to define "scientific method" in a way that includes all scientific methods and doesn't include other stuff. So keep in mind that the following general definition has its limitations! We can describe the scientific method as:

1. Formulate a hypothesis.

2. Test the hypothesis through experiment or observation.

3. Accept the hypothesis as (provisionally) correct, discard the hypothesis, or modify the hypothesis.

This describes the process of improving and developing technology, long before the Scientific Revolution. For agriculture, one might notice that barley planted where there had recently been a crop of peas grew well. One tries it on a larger scale, to confirm or refute the hypothesis that alternating crops of grain and legumes gives better grain yields. It works, and crop rotation is developed. There is a strong motivation to test the hypothesis - the direct economic benefit of higher yields. Other technologies also offer straightforward testing of hypotheses - can you dig more easily with your new and improved shovel design or not?

There are differences between this pre-scientific application of scientific method to technology and the ideal modern application of scientific method:

1. First, we have a much better understanding of testing hypotheses, including things such as experimental design (e.g., using control groups, using double-blind methods where placebo effects or experimenter bias matters, etc.) and the statistical analysis of results. Without a good understanding of testing hypotheses, it's possible to come to the conclusion that some things work, when in reality they don't work. For example, one can conclude, through testing without controls and without proper statistical analysis of the results, that rain-making magic works. This isn't a major issue in technology when it comes to functional changes, since it is often clear if the function of the object is improved (but not always - one might still decide that racing stripes make cars go faster).

2. Second, we can now draw on a much broader range of knowledge for developing our hypotheses. Modern science can be very useful for the improvement of technology. One finds many claims that such-and-such scientific discovery/principle/theory is essential for some particular technology where this isn't the case, but these overly strong claims of science being essential don't mean that science is never useful for various technologies. It often is. This, together with improvements in technology providing scientific insight and thus advancing science, has led to relatively rapid development of science and technology side-by-side. When technology largely depending on parent-child and master-apprentice transmission, often without the use of written material, the amount of "science" that could drive improvements was much smaller. When scientists largely ignored technology (science as a science (book knowledge), and technology as an art (practical skills)), science had less input and less improvement. One key advance is stronger linking of theory with technological improvement. If there is no known theoretical reason why some change should improve the performance of the technology, more effort is made to develop improved theory if the change passes the testing of the hypothesis. In the absence of suitable theory, some hypotheses are subjected to a stronger evaluation. Thus, one is less likely to conclude that racing stripes make cars go faster, and if testing shows that they make cars go faster, finding out why this is so is recognised as an important question.

3. Third, the application of science to technology allows the use of quantitative scientific models in the service of technology. The potential performance of objects in their intended purpose can be calculated using mathematical models, and the design optimised in this way. This is an important component of modern engineering. Before suitable quantitative models in science, and their application to technology, technological progress was largely through cut-and-try - a new design would be made, and then tested. A key difference between cut-and-try and the use of quantitative models is that quantitative models allow the expected improvement to be quantitatively predicted. This is a great aid to testing the hypothesis - the question stops being simple "Is there an improvement in performance?" and becomes "Is the improvement in performance the same as our prediction?".

But these are improvements in the method, rather than a distinct method [1]. The hypothesis-test sequence can be carried out, with resulting improvements in technology with tests less rigorous than we would prefer today, without the use of quantitative models, and with less input from science.

Since the scientific method was in use long before the Scientific Revolution, it can be reasonably suggested that a key element of the Scientific Revolution was taking the hypothesis-test method from technology and applying it to science. A central concept in this process is the idea that scientific hypotheses can be tested. Rather than science being about description and classification, and story-telling explanation, it should be about quantitative predictions (i.e., mathematical models) - this is a the essential difference between Newtonian science (quantitative models) and Cartesian science (stories), and is what led to the enthronement of physics as THE science (as per Rutherford's (alleged) statement that "All science is either physics or stamp collecting"). The importance of quantitative predictions in the scientific method as applied to science is the improvement in hypothesis testing - does the quantitative prediction match the measurement?

The importance of hypothesis testing in science can be seen from the advances in Medieval science in the 12th/13th century scientific revolution. This resulted, in part, from new methods for better testing of hypotheses - the recursive argument method. This wasn't a method for experimental testing, but allowed improved testing of ideas through oral and literary disputation. For more on this, see Beckwith (2012).

Thus, better methods for the testing of hypotheses, together with increased awareness of the importance of testing hypotheses, led to more rapid advances in science (12th/13 century scientific revolution, the 17th century Scientific Revolution, and the modern acceleration of science). Meanwhile, the testing of hypothesis had long been central to the improvement of technology - the question "Does it work better?" occurs quite naturally in technology.

Footnote:

[1] That is, unless we define the scientific method in such a way that quantitative models are required. But this definition would exclude much modern science, and would therefore be a rather dubious definition.

Reference:

Christopher I. Beckwith, Warriors of the Cloisters: The Central Asian Origins of Science in the Medieval World, Princeton University Press, 2012.