Changing Out a "Vintage" Dimmer Switch
You have a "single-pole" arrangement, i.e. one switch controlling the load (the light). You simply need to turn off the circuit breaker controlling this circuit, remove the two wires from your existing dimmer switch and place them on the two wire terminals on the new dimmer switch (the terminals that are NOT green). It does not matter which wire goes to which terminal.

There does appear to be a ground wire (bare) under a screw in the back of the box; if possible you should attach a short green or bare wire from the green terminal on the new dimmer to that same screw (or alternate ground screw) in the back of the box. This is not required for the new switch to work; it is a safety feature. 

Attach the dimmer and cover plate to the box, then turn the circuit breaker on. Done.