Joycon stick stuck and broke inside

Grasp hold of the circuit board and at the same time press the back release button.

Slide the piece attached to the circuit board up the same as you would to release the entire joycon normally.

That should get it off the main body of the Switch. After that you will need to at a minimum replace the slider rail on the joystick. It's supposed to have two ribbon cables attached to it; one goes to the connector at the bottom (that one looks like it might still be usable on yours), and it has one more that goes to the circuit board you just took of. That one looks like it's missing; I'm guessing it was torn off and is still attached to the joycon's motherboard.

There's an iFixit guide for the left side rail replacement, but I didn't have any luck locating one for the right side. I'd suggest following the left side guide as it should be close to a mirror of the right side, but use common sense if you run into something that doesn't match.

Left Joy-Con Charging Rail Replacement - iFixit Repair Guide

Here's an example of the part you'll need; be sure any replacement you buy comes with both ribbon cables; otherwise it'll be worthless to you.

Original Replacement Joycon Left Right Slider Rail With Sensor For Nintendo Switch Joycon Controller With Flex Cable - Accessories - AliExpress

I'm guessing there's some broken plastic in there as well, so you'll either have to glue the broken pieces back together or else do a shell replacement on that joycon.

Nintendo Switch Right Joy-Con Shell Replacement - iFixit Repair Guide