I see a lot of words starting with 引き, for example 引き下げる, 引き出す.
引く has many meanings but I think in these cases it means 'to pull'.
My question is, for example in 引き下げる, what is the purpose of 引き? How does 引き下げる differ from 下げる?
引き下げる ('pull' + 'lower') is just that much more illustrative and specific than plain 下げる('lower'). It doesn't mean just any old kind of lowering, or pushing down or beating down for that matter, but pulling down.
It has, of course, figurative uses (as in 値段を引き下げる) where the action being described doesn't involve specific physical movement suggested by the letters of the word (like cutting down prices doesn't involve physical cutting or spatial lowering as cutting down trees does).