It wasn't always this way. When it was first built, it was called the South West Urban Motorway, connecting the M62 at Gildersome to central Leeds, where it terminated at what is now junction 3. The M1 entered Leeds as the South East Urban Motorway and terminated at the same junction. There, both routes collided, dumping all their traffic - five lanes of it - onto what was basically a big roundabout. The M1 did so by narrowing from three lanes to one (the way forward blocked by a row of permanent cones) and then travelling around a very tight one-lane loop. The original plan had been for the motorways to merge, to form the never-built North East Urban Motorway, and that would have crossed the river to distribute traffic across the city.
In 1999, with those plans long dead, the M1 was diverted onto a new course away to the east of Leeds to terminate on the A1(M), vacating the South East Urban Motorway. Some new connections were built at junction 3 in the middle of Leeds, connecting up the South East and South West Urban Motorways, and some renumbering work was done. The M621 now enters from the M62 as it always has done, swerves sharply to avoid the city centre, and heads out again to the south on what was once the M1.
The result is that the M621 forms not just one of the highest-capacity and most important approaches to Leeds city centre, but two of them. Both its south-eastern and south-western legs offer high capacity, effective ways in, and new connections provided since 1999 mean that not only is junction 3 better at handling the traffic deposited there, but there are now other ways into the centre as well.
One of the most unusual things about this road - and perhaps the reason that some people will have ended up on it without meaning to - is its junction with the M1 at Belle Isle. M1 junction 43 sees the M621 take priority through the junction, flowing directly into the M1 to the south - so if you travel north on the M1 and don't watch the road signs, you'll find yourself on the M621 through central Leeds instead of continuing north. As with any urban motorway, keeping an eye on the signs is always advisable!