I can think of a few reasons.

First, there is a school of thought that says that the planet is ultimately doomed, and if we want to survive as a species, we need to get off of it. The first step is setting up an off-world base and maintaining it. The Moon is the most convenient place for that.

But from there, we also need to get to other places in the universe. The gravity on the moon is significantly weaker than Earth, and it would cost much less fuel to launch to other planets (and beyond) from the moon than from Earth. Of course, that advantage is negated if all that fuel needs to come from Earth, so the space nerds would need to learn how to make rocket fuel out of lunar materials, which they are working on. (A moon base could also allow us to mine passing asteroids for raw materials with much less fuel.)

There are also definite scientific benefits. We are currently launching remote observation satellites from Earth into the Lagrange points so they can point outward and be shielded from emissions from Earth. But we can put a much larger manned apparatus on the dark side of the moon and get all sorts of data that those remote telescopes alone can’t gather.

Finally, there is the simple competition between nations. The US will try and establish a moon base first, to try and call “dibs”. Eventually, we may end up sharing scientific endeavors there, like we “share” Antarctica. (Actually, the Antarctic situation may be a good analogy for how we will eventually manage the political situation on the moon…)