I've never heard anyone say things like this, but it seems like it's a version of "I'm on it". But it completely depends on how the line was said.
In any case, if the line was said the "I'm on it" sense, on means "engaged in" or "in the course of," again depending on context.. For example if you had the sentence:
Sally is out on errands.
It could be reworded as:
I'll go do the errands.
Oh, don't worry, Sally's on it.
Meaning, of course, that Sally is engaged in/doing the errands. Or...
Who's doing the errands?
Don't worry, I'm on it.
Meaning that you are in the course of doing the errands.
If you're volunteering for work, I could imagine one saying "I'm on [action]," though again, I don't usually hear people say things like that. But I imagine if you were watching something like Scandal someone might volunteer to do something they're good at by saying "I'm on [action]." Or in this way:
We need a person to do the dishes, a person to mop the floors, a person to to inventory, and a person to wash the windows.
I'm on washing the windows!
But on could also mean "having [the thing in question] as a focus". So if you told a friend something like:
I'm on a never-ending quest to save my girlfriend!
Then sometime later, they might ask (if your girlfriend isn't saved yet):
Hey, are you still on saving your girlfriend?
Meaning "is saving your girlfriend still your goal" in this case. Especially if you don't seem to be right in the middle of saving said girlfriend but doing something marginal or unrelated.
You're still here? Are you still on saving your girlfriend? You know you're not going to find her here, right?