Installed dimmer switch caused another switch to stop working
You really need a multi-meter or a non-contact voltage tester to find out which wires are hot, but if you want to you could try to do this without testing.

The white wires are "neutrals" (not grounds). Looks like you have bare grounds connected to the metal box with screws in the back. Leave those alone.

Do the B, R and W wires in the right cable go to the light in your room? If so, the B &amp; W wires in the cable to the left are the feed to the hall and the other bedroom. The B is hot and the W is neutral. Somehow you disconnected the feed and did not reconnect it.

Probably the B and the R on the right were connected to the original switch. If so, one or the other of the B and R is always hot and the other is the switched hot which controls the light in your room.

The B in the left hand cable should be connected to the always hot from the right cable. The two W wires should be connected to each other (and to nothing else).