Generally-speaking in recent times, it has been done by taking animal models (usually mice, as they reproduce fast and actually share the vast majority of their genome with us), and creating “knockout” forms of them where one or more stretches of DNA that code for a protein (which is actually all that a gene is) are “switched off”, and then seeing how this effects the animal.

I actually once got to meet Mario Capecchi, the researcher who pioneered “knockout mice”, and won a Nobel Prize for it.

Prior to this innovation (which I believe only came in the 2000s, although maybe it’s a bit older), biologists were mostly in the dark about which genes related to which traits, and it was really mostly guesswork. Genetics used to focus more on Mendelian heredity (which if you ever learned any biology in school you were likely introduced to), which connects observably inherited traits with a sort of theoretical entity called an “allele” (which don’t necessarily correlate to genes), and then attempts to sort of reverse engineer the rules for how that trait is passed on (whether it is recessive or dominant, etc.). Since most traits actually have more than one gene involved (since literally all a gene does is act as a molecular blueprint for a single protein, that’s it), sometimes in fairly complex ways, this way of looking at genetics is obviously very imprecise and rudimentary, and it’s mind-boggling how much genetics has advanced just in the past twenty years (it really is almost like going from abacuses to electronic supercomputers in just a couple decades). Thirty years ago we hadn’t even sequenced a whole human genome, now we can pin down certain traits to particular stretches of DNA, and then even selectively alter that DNA if we want (or put it in other organisms, etc.).

EDIT: There are other methods as well, but they tend to be more indirect and imprecise. Like mutant forms of some simpler organisms like C. elegans roundworms and E. coli bacteria have been studied and selectively bred for a while to try to disentangle which stretches of their DNA the mutations might lay in (as with bacteria especially there’s generally less DNA to look at, so this is more feasible than with organisms that have huge genomes like us, or surprisingly like a lot of plants; seriously, plants are so much more genetically-complex than you might expect, in part because a lot of them require a ton of different enzymes for complex biosynthetic pathways of various organic chemicals, and enzymes are proteins, and therefore each is tied to a particular gene).