Why is low-dimensional discrete representation of input space useful? Self-organizing maps undergo unsupervised training to produce a low-dimensional, discrete representation of the input space. Why is this useful?

I think the results of the self-organizing map would have to be combined with some other machine learning algorithm or neural net in order to solve problems. Is this the case? If so, what algorithms or neural networks would benefit from being complemented with a self-organizing map? If not, what problems does the output of a self-organizing map solve?
It may or may not be useful, depending upon the circumstances.

In some cases it can be useful as a visualization technique.  The representation is low-dimensional, and often 2-dimensional -- which means it can be visualized.  That said, there are many other methods of representing data in a low-dimensional space so it can be visualized, and it's not clear that self-organizing maps are the best way to do that.

Sometimes self-organizing maps are used as a general form of clustering, and can be useful anywhere that (unsupervised) clustering is useful.

That said, I agree with your general reaction.  Often, self-organizing maps aren't very useful.  I would not expect neural networks or other algorithms to benefit from being combined with self-organizing maps.  Self-organizing maps don't really solve any clearly defined problem, and probably aren't of all that much use in many/most practical situations.

If you had never heard of self-organizing maps, you probably wouldn't be missing much (in my personal opinion).