Anyone know the origins of this phrase?
I’ll make a guess at it.
“Does the vessel of your mind intercept and follow the path of the vessel of my dissertation?”
That is, “Do you follow me?”
I don’t have a cite for you, but I’ll take a stab at this one, so to speak 
It’s of nautical origin. A drift is a tool that a sailor would use to knock out the pins holding the lines to the sail in place. You’d only need to employ a drift in response to new orders from the captain. “Yarrrrgggh, go thataway matey!” Sailor 1 changes the rigging he’s responsible for and tosses the drift to sailor 2, who then performs his tasks. They’re both operating under the same orders, and presumably have the same comprehension of what’s required to follow through on them. So, there’s your origin: “catch my drift”
That’s mostly a WAG, but it makes sense to me. If someone more knowledgeable than this landlubber can come up with a cite or a better explanation, please do!
Probably of Biblical derivations – from John 3, when Jesus is talking to Nicodemus.
Jesus tells Nick that you must be born again – which Nick, taking him literally, says is impossible. After a little more explanation, Jesus tells him:
The actually English phrase in probably of more modern coinage, but the basic linking of speaking in non-literalist terms to the wind goes back at least this far.
I vote for usage in the Western US in and around the 1860’s-1880’s.
Of course, drifting is a method of fishing, as well. I haven’t found a clear citation for its origin, yet.
Here’s my theory:
I can find citatation of “catch” meaning to grasp the meaning of, to understand. They are from 1881 and 1882. Both say “catch the idea?” They are both cited in my Lighter, American Slang.
I always have felt that the phrase “catch my drift” as I have heard it used since the '50s or '60s, I associate with Westerns or Gangster movies. Less educated types, dontchaknow. I can still see a bad guy in a black hat sneering into the camera on Saturday morning <catch my drift?>
And the verb/noun “drift” was certainly a used term in mid-1800’s Western life. It had meaning about cattle, etc.
So, I just figure the time-frame for the citations of “catch” would complement a Western US origin.
I, too, have not found a citation for the actual phrase, as tom said.