The common formula "by or on behalf of X" is, I think, fairly transparent: it means "by X or on behalf of X".
But as I'm sure you know, we don't say "on behalf of me"; rather, for short/light values of X, "on behalf of X" tends to become "on X's behalf".
So "by or on my behalf" is clearly intended to mean "by or on behalf of X" with X being "me".
You only asked about the meaning, but I'm guessing you're interested in the origin as well. I don't know, but I'll do my best . . .
The linguist Neal Whitman has documented a whole bunch of instances of what he's dubbed "right-node wrapping", whereby a construction like "A and B C D" means "A C and B C D". For example, the chorus of the song "Friends in Low Places" contains the clause "the whiskey drowns and the beer chases my blues away", meaning, "the whiskey drowns my blues and the beer chases my blues away"; so the "my blues" part is an object of both "drowns" and "chases", even though it comes right in the middle of "chases ____ away". (The name "right-node wrapping" is because the coordination has two partial-verb-phrase "nodes" — "the whiskey drowns" and "the beer chases away" — and the right node "wraps" around the shared direct object.)
So one possibility is that "by or on my behalf" is a more-extreme variation of right-node wrapping, where my somehow gets implicitly split into me + -'s, and then the node on -'s behalf wraps around the shared object me.
Another possibility is "by or on my behalf" originated as an editing error (someone started with a document that had, say, "by or on behalf of the undersigned", then changed "the undersigned" to "me" and "on behalf of me" to "on my behalf", without noticing that "by or on my behalf" didn't make as much sense) and then simply took root.
Either way, lawyers are very big on copy-and-paste (generally preferring to reuse language that's stood the test of time and court, rather than drafting their own wording that may get on a judge's bad side), so it's possible that many of the documents that use this phrase were not by writers who would use the phrase themselves.
And also either way, the relevant lawyers would probably be familiar enough with "by or on behalf of X" that even if they noticed that "by or on my behalf" sounded strange, they'd probably have no difficulty recognizing the formula and interpreting its meaning correctly. They probably wouldn't notice (or care?) that it's very opaque to the rest of us.
