How did the astronauts film themselves leaving the moon

Did they leave a camera there? If so, why?

They did not film themselves. The liftoff of Apollo 15, 16 and 17 from the Moon was broadcast live using a TV camera mounted on the lunar roving vehicle. This camera was operated remotely from Earth by a guy called Ed Fendell in Mission Control.

Mounting a camera on the rover was originally done because the rover would allow the astronauts to travel a few miles from their landing point, so in order to be able to keep sending back TV they had to take the camera with them. It was realised pretty soon that this allowed the opportunity to broadcast the moment of liftoff if they just parked the rover a little distance from the LM and aimed the camera at it. There was no need to bring the camera back. It would just have been dead weight.

I guess you're referring to the videos of the launches of the Apollo 15, 16 and 17 lunar modules from the surface. There was a video camera mounted on the lunar rover, which they parked in a position so that it could film the takeoff. It was pre-programmed to pan up at a given rate and watch the module ascend.

Did they leave a camera there?
Yes

If so, why?
So they could film themselves leaving the moon.

Basically, NASA via the US government spent a gigantic amount of tax payers money on the Apollo missions. So they made sure the public could see where their money went by filming and photographing just about everything they could.

It was done remotely, and yes the camera was left behind.

I'm the one filmed them...

Bipolar disorder. Split personality.

They for got the one that had the camera had to be picked up before they left the moon
.

In the news footage? If look closely you'll see the word "SIMULATION" on those.