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Can someone explain to me the reasoning behind the change of clef in those measures? Without knowing anything more about it than having just looked over the score, that part appears to be easiest to play (or at least playable) with the right hand alone -- what's the objective of playing those notes using the left hand instead?

Hi Frank, I think you will see why pretty easily when you sit to play it through...the left hand E notes kind of " chase" the right hand ones up the keyboard...much smoother with both hands! Hope you get inspired to learn it

I think Frank was referring to the alternating D# / E notes, which are elsewhere in the piece played with the RH only, but in this case the LH/RH alternate for several D# / E pairs.

And for example, IIRC (can't check right now), my Henle edition also has this alternating LH/RH pattern in measures 38-40. But other editions play those notes in 38-40 with the RH only, but also use the LH/RH alternating pattern in 13-15. So why the difference?

An explanation might be, that in 13-15, after playing the high E with the RH, it is easier to start playing the D# / E pattern with the LH, otherwise you have to jump back with the RH quite quickly (and this does not apply to 38-40, so maybe that's why some editions notate this part in 38-40 for RH only).
But if you start the D# / E with LH, then at some point you have to switch from LH to RH again.
Personally, I don't have an explanation why this switch was notated in this fashion (i.e. switching back to LH once more after only 2 notes in RH, before then finally switching to RH for good). Maybe just because it looks more "virtuousic"?
I certainly feel more "virtuousic" when I play them in this fashion.

I asked this same question of my teacher (about the alternating LH/RH notes where really you only need one hand), and she told me that it is played this way because the little bagatelle we're discussing is all about communication between the hands. The melody moves seamlessly between them for most of the piece, tossed lightly back and forth from one to the other, and in the spirit of that communication this section is shared between them as well.

I don't know how true or accurate this explanation is but I do feel that it fits well with my idea of the piece. I play it as written on the occasions that I recall how lovely it is and pick it up again.