While it may seem trivial, I think good edit summaries are extremely important. For new users, they help them understand their mistakes and learn from them. For editors, they are a kind of "rubber duck problem solving": Having to explain the purpose of the edit may make the person think more carefully about their edit, and this may in turn cause them to think of ways to make even better edits.
The problem is that I only understood the value of good edit summaries after I got enough rep to review proposed edits. After wading through a few dozen edit proposals (often from new users) with useless summaries like "changed a few things" (great, now I have to carefully read the two-screen long question, twice, to figure out what you were trying to do), contrasted by some from experienced users like "the question was not clear on point X, so I edited based on reason Y and Z to make it clearer, as stated in a comment by user U" I very quickly realized that edit summaries matter, and new users are often very bad at writing them.
I can't really blame the new users, because before I could review edits, I was exactly the same. Looking back, out of the dozens of edits I proposed, I don't think I wrote one summary that wasn't utter rubbish. But, I just didn't know any better. Now that I've seen exactly what sort of difference it makes, my opinion changed completely, and I can't even bring myself to suggest an edit for a question unless I would be able to write a good, coherent summary for it.
The 10 character minimum is certainly a great first step, but it's inadequate. Clearly, it failed for me. As a new user, I understood in principle why the limit was there (and more generally, I understood why good commit messages are important from what I knew about versioning code), but I was under the impression that somehow it didn't apply to my edits - I would just put in throwaway summaries like "rewrote question" to make it shut up and let me submit.
It was only after being on the receiving end of a summary, so to speak, that I could really "get" it.
Increasing the character limit is probably not the way to do it, because length isn't the point, and you could create problems for some legitimate but short summaries.
I'm also pessimistic about mere explanations. As I said, it was already clear to me that poorly explained edits are bad. But I was under the misapprehension that my edits are so minor that they are self-explanatory, and thus an exception; only after reviewing edits could I understand that truly minor (and self-explanatory) edits are extremely rare - just because you changed a few characters doesn't mean the whole post doesn't need to be read to judge the validity of the edit.
Since the most effective lesson was experience, perhaps we could somehow allow even new users to participate in review? There are good reasons for a reputation limit on reviewing. But maybe new users could review edits and only provide votes to help the 2k+ reviewers who actually accept or reject a proposal?
You could argue that we should do nothing. The edit queue on SO is usually empty, and new proposals are cleared very quickly. But, even so, making the process more painless to reviewers might help the reviewers do a better job (going by the logic that tedious tasks are harder to do well than effortless ones).
Another possible option I can think of is to collect some known "bad edit summaries" like "changed some things", check if someone attempts to write a summary similar to these, and warn the user if it looks like they've written a bad summary.