Help replacing old fan switch
The black wires from switch to switch are power jumpers. The power is fed to the upper switch by the black wire on the top left. The jumpers then carry the power to the other two switches. The switches have an internal connection between the two terminals on the top edge of each switch.

The wires in the lower left terminal of each switch is the switched power to the each fixture.

In your new combination switch unit, there is a common power feed, eliminating the need for jumpers to the three individual switches. They are connected internally. You connect the incoming power, the black wire on the top left of the old switch, to the common. Then the three output wires from the old switches go to the three terminals on the right.

Many older switches lack a grounding terminal, which is now required on most new switches. If you have a ground in the box, it should be connected to the ground terminal of the new switch.

The new switch may or may not solve the don't work problem, but it's a good place to start. If it doesn't solve the problem, you need to find out what else is disconnected or non-functional.

Note: The incoming power wire looks like it might be 12 gauge. If so, you can't use the push-in connectors (they only work with 14 gauge), but must wrap the wire around the screw terminals (clockwise). Most practitioners consider this to be a better connection than push-ins, so you may wish to do this regardless of wire size.