This is a very imprecise, general question. Mostly because I'm not exactly sure what I'm after. I just think that I miss something crucial here.
In the context of model categories, homotopy theory, derived geometry and operads ect., it is often a major step to replace a structure, an object or something with a (co)fibrant replacement.
What do we achieve by this? Why is it "better" to work with these replacements, once we add any notion of "homotopy" into our theory? (I know there is no precise meaning of the word "better" in this context, I just don't see the reason, why people put so much thought into these replacements).