"do not let your left hand know what your right hand is doing"
The reason I think that, is because in ancient Roman times they had a way of counting on your fingers where you could count up to 100 on your left hand alone (then you could use your right hand after that for even larger values). [1]. So if the context is "small values" and charitable giving, it makes more sense to me at least.
"do not let your left hand know what your right hand is doing, so that your giving may be in secret."
It's just hyperbolic/poetic language. It's not possible to do something in such a way that your right hand doesn't know what your left hand is doing, because they are both connected centrally (and they can't know anything to begin with).
"Be so secretive that your resting hand doesn't know what your charitable hand is doing."