I'm not really sure how democrats think the Russians could "rig" an American election.
The same way that Americans or any other foreign power could rig American elections: Hack the electronic voting machines. It's demonstrably easy to do by anyone with the right skills, and would likely be done by anybody with the desire to see one candidate prevail over the other. The machines are hackable, and we know that the Russians would likely hack them if they could, so it's reasonable to consider that they did. What would stop them?
Watch: Hackers Demonstrate How to Crack Into Electronic Voting Machines in Minutes
Sounds more like a conspiracy theory to me.
It is a conspiracy theory - or more properly, a conspiracy hypothesis - and probably an accurate one. There is also statistical evidence that the voting machines were hacked: Rigged election: Donald Trump won every surprise swing state by the same 1% margin
Every crime or other unethical act ever committed by two or more people planning together in private is a conspiracy. If you rob a liquor store alone, it's not a conspiracy. If you talk somebody into assisting you, perhaps even just by waiting out in the parking lot with the car running, it's a conspiracy.
Remember the Hillside Strangler? Turned out to be a conspiracy. There were two of them working in tandem, Kenneth Bianchi and Angelo Buono. That's a conspiracy, and the prosecutor presented his conspiracy theory to a jury.
Conspiracy theory is a phrase used to demean an idea and its source. It implies that they are both lunatic fringe.
Note that there are two competing conspiracy theories regarding 9/11, but only one is called that - the one implicating American involvement. The official story is that Arabs conspired to commit acts of terrorism by themselves. That would also be a conspiracy, but not one considered crackpot.