How to on this retrofit led recessed light

Depending on the not yet clear "what's going on with the old wiring" - somewhere, the old wiring ends up feeding power to the old lights.

The retrofit LEDs I've used have a socket-to-screw-in-base adapter included, and I just cut off the screw-in base to get the mating connector that matches the new fixture, and wire it to the wires feeding power to the old lights, where there isn't a socket to screw that into. If there is an old socket that's not in the way of the new light, (or which can be raised out of the way in the old can) just use that adapter as intended.

Alternatively, you could (perhaps, not having seen that half yet) cut the mating plug off the old fixture and wire it to the new fixture. I generally choose the newer part to "save" in this situation rather than the older one, though.

You often need to attach a separate ground conductor to the old can from the new light, as well. The old lights often don't have this, IME.

In this type of installation, you fit the box, then you reach through the hole and wire the box, then you install the lamp innards into the hole.

You removed too much of the innards.

Put it back and see if you can remove part of it in a way that makes sense. (feel free to do this on the bench, no need to strain your neck).

Also, some of these newer type lamps are fluorescent, and have ballasts that sit up inside the empty box in your first photo. In those cases, the wiring from the box to the innards may not be mains voltage, it may be ballasted power intended for a specific CFL tube.