Lacking a canonical explanation, here is what I always thought. Yes, it considers the canonical pseudo-explanation "because magic".
Where does the water go when it falls?
And for that matter, why does it even fall? We must assume that the Discworld has gravity , but where does it come from? From the balance of the Disc on top of 4 elefants on top of a turtle that travels through space I state that Great A'Tuin (the turtle) is the main source of gravity. So the direction to the turtle is "down". So when the water falls off the rim of the disc it does so to go towards the turtle. That makes sense. Chelonii and water are two things that just make sense together.
But the waterfall wouldn't stay as a waterfall for long. After leaving the pressure of the atmosphere, the water would separate in billions of very tiny rain drops. Because that's just what water does in low pressure . That's why the waterfalls seem to disappear. The water is just not visible anymore, but it's there, all around Great A'Tuin, who swims in the thinest sea ever conceived.
How does it come back to the Disc?
When the sun goes under the Disc, it heats up all that water around A'Tuin and the steam goes up again. "Up" is the opposite of down , so it would go away from the turtle. Why back to the Disc, of all directions? Pseudo-theories are welcome in the comments, but I think it's probably because the magic of the disc attracts it (you know, because magic and steam/smoke are things that also make sense together, have you never seen a magic show?).
Once back in the atmosphere (atmosdisc??), it tends to go towards the Hub. Why? Again, because magic. The Hub is stupidly cold because the magic field is so thick that the light can barely touch it. Also, because of the ice giants (or is it the other way around? Cause and effect are funny stuff). On its way to the Hub it will meet cold opposing cold streams and geography that it will make it fall as rain again. Probably on Ramtop mountains. Definitely not in XXXX. And from there, the cycle starts anew.
Why can't we just say "It's magic"?
But we just did. What I just proposed would be consistent with Ponder Stibbons statement:
Without magic the seas would run dry.
a) Without magic, the water wouldn't come back to the Disc and would eventually run out.
b) Even if the water did come back, the Hub wouldn't be so cold. The water would stay as steam, heated up by an unrelenting sun. The temperature of the disc would skyrocket. Life would mostly go extinct. And yesh, the seas would (practically) run dry.