"Dumb question, but I confess I dunno UK specifics."
Not at all dumb and deserves an answer.
The leader of the largest party after an election is normally the one asked to form a government. If that party has an overall majority that's pretty well a shoo-in. If not then someone has to try to form a coalition or even run without a coalition or overall majority.
It's possible for a party with no formal coalition or no overall majority to govern; they might not get all their policies through and can be displaced by a vote of no confidence - that happened to Callaghan when he lost the support of the Ulster Unionists. In general that's not a good form of government because it's apt to hand disproportionate power to a very small fraction of the electorate.
There's an extra twist when there's a no-overall majority in that the previous governing party might try to form a government even if they're not the biggest party so everybody has to wait until they decide it's a non-starter which caused some hesitation after the 2010 election.
Once a party is in government they generally have the option to change leader and hence PM without having to call an election. That could be through ill-health (Eden and Wilson, for instance, within living memory) or by revolt (Thatcher), because the PM sees the runaway train coming down the track (Blair who handed over to Brown just in time for the latter to catch the consequences of his 10 years of pretending that housing costs could be ignored when measuring inflation, even if that meant ignoring rampant house price inflation /rant). It could also come from an intention to hand over to a new leader who'd then have time to establish himself before a general election which was Cameron's stated intent although he might have intended to leave it a little later.
During the course of a parliament a government can lose its majority by by-elections which, IIRC, was what happened to Callaghan, and then becomes victim to a vote of no confidence.
The current situation has the extra factor that there's an Act in place which imposes a strict 5-year interval on general elections. That was part of the previous coalition arrangement; it protected the LibDems against a snap election and also limited the Tory right wing's ability to rock the vote. It could, however, be repealed any time if Parliament wanted and presumably the Queen could, acting with the Privy Council if necessary, ignore it in an emergency.
An incoming Remain PM could have the option of asking Parliament to repeal that act and hold a general election fought primarily on that issue so that it would be a new Parliament that would (or wouldn't) invoke section 50; there seems to be a general feeling over the last few decades that kicking stuff to the other side of a general election is a Good Thing in that it gives the electorate a chance to reflect. Irrespective of the next PM's views it would still be a Good Thing. In fact I think even some of the Leavers were saying that - or it might just have been Boris.
HTH