What could be causing my garage door to randomly open and close?
First rule of electricial/electronics is cycle power to see if that solves the problem. Unplug it - give it a few minutes and then plug it back in. If the random openings stop, the unit just needed a hard reset.

Next Up - Disconnect the low voltage/control wires from the connector at the opener. If the door still opens randomly then the problem is in the opener itself.

If the random openings stop, hook the low voltage cables back up to the opener.

If the random openings start again, the problem is in the control side of things.

Remove and replace as needed.