Why do Topologies get "finer"? Why are topologies with many elements called "fine" and topologies with few elements called "coarse"? It seems as though the finer a topology is, the more likely it is for a function defined from that topology to be continuous, and conversely with coarse ones - for example, every function from the discrete topology is continuous, and every function to the coarse topology is continuous. Is there some intuition that explains this choice of words?

I think of it in terms of resolution: in a finer topology, the open sets "distinguish points more". For instance, fewer sequences (or nets) converge, and fewer functions with the finer space as the codomain are continuous. This is directly because points are more distinguished from one another. On the other side, more functions with the finer space as the domain are continuous, because the requirement "close points get mapped to close values" has to be checked on fewer points in the domain (again because points are more distinguished from one another).