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).
Imagine that you're looking at the underlying set $X$ through a visual filter that limits the resolution and makes everything blurry. If that filter is very coarse, then more nets in $X$ will appear to converge to certain points, since everything is blurry and you can't really tell that there's a gap there. On the other hand, if its a very fine, high-resolution filter, then fewer nets in $X$ will appear to converge, because you've got the resolution to tell that they aren't really approaching. The highest resolution filter is the discrete topology; here, the only way a net can converge to a point $x$ is to eventually equal $x$. Otherwise, you'll have the resolution to be able to see that it isn't really converging!
I fully admit that this is a moderately dodgy viewpoint.
I interpret "finer" as "finer control over building open sets". Think of topologies as building blocks. If $\tau_1 \supseteq \tau_2$, then $\tau_1$ has more building blocks, and finer control over what can be built with them.
I don't think the use of "fine" and "coarse" has any deep meaning.
Picture a boulder. That's the trivial topology, the coarsest of all.
Now take a gigantic sledge hammer and give the boulder one hard smash. The boulder breaks into a rather course collection of big rocks. Those big rocks generate a somewhat finer but still rather course topology.
Now bring in a whole bunch of sledge hammers and smash those big rocks over and over, breaking them into a somewhat finer collection of smaller rocks. Those smaller rocks generate a somewhat finer topology.
Now spread those smaller rocks over a large, hard, flat surface and run over them a lot of times with a steam roller. The steamroller breaks them into a much finer collection of pebbles. Those pebbles generate a much finer topology.
Finally, put those pebbles in the bottom of a tank and run water over them vigorously, back and forth, for a few years, maybe a few centuries, perhaps a millenium or two. Now the stones have broken up into very fine grains of sand. Those grains generate a very fine topology.