Great question; thank you for raising your point. I believe that the when the claim is made that science is not value-free various things may be involved. There might be at least four points to consider.
First, there is the thesis that the very practice of philosophy itself involves values --minimally, given a realist view of truth (which I hold and I think you do too, e.g. water is H20 if true if and only if water is H20)-- a commitment to discovering the truth about various phenomena, being reliable or trust-worthy in recording observations and constructing hypotheses, theories, reporting anomalies given such and such a theory, and so on. Perhaps this is not radical news, for it seems that virtually any social interaction in which we trust each others' reports / testimony, all sorts of values and commitments are relevant. But what might be added are three other points.
Second, the practice of science itself is guided in light of what scientists (or those who support the practice of science) believe to be valuable (or worthy of scientific inquiry).
In addition to that (number three) Thomas Kuhn, a prominent 20th century historian and philosopher of science, famously argued that scientific progress is often suffused with matters of value and subjectivity when individual scientists choose to retain normal science (as practiced in conventional, institutionally entrenched labs, classrooms, institutions, etc) or seek to bring about a scientific revolution (e.g. as we find with Galileo, Newton, Darwin, Einstein...). All this is compatible with realism and your initial point that the atomic structure of water (for example) does not depend upon political or social contexts or the personal subjectivity of observers.
Fourth, perhaps physics and chemistry (which involve the water example) seem straight forward facts as distinct from values, but when we move in the direction of biology, values of various sorts come into play. This might be especially apparent when scientists employ terms like 'health' and 'disease' --whether in terms of organisms or ecosystems. The health of an organism seems the equivalent to the notion that the organism has some good or can be in a (for it) good state. It is also not uncommon for philosophers of science to contend that in the social sciences various concepts are social constructions (e.g. the diagnosis ADHD) and reflect more our values than "the facts."
I hope that is a helpful beginning. The philosophy of science entry in the Stanford Encyclopedia of Philosophy is very helpful.