Learn Russian: Собака отдыхает на диване.

Because it is describing a location. When you say where something is located (using в for 'in' or на for 'on'), the noun usually takes the Prepositional case. For most masculine nouns ending in a consonant like диван (sofa), you just add an -е to the end, making it на диване.

Why do we use на instead of в for 'on the sofa'?

In Russian, на means 'on' (on a surface) and в means 'in' (inside). Since the dog is resting on top of the cushions, we use на. If you said в диване, it would literally mean the dog is somehow inside the actual furniture!

Does отдыхает mean 'rests' or 'is resting'?

It means both! Russian does not have a separate continuous tense like English does with 'is resting'. The single word отдыхает covers both 'rests' as a general habit and 'is resting' right now. You just use the context to figure out which one sounds best in English.

I don't see words for 'the' in this sentence. How do we know it is 'the dog' and not 'a dog'?

Russian does not have articles like 'a' or 'the'. The word собака can mean either one. We translate it as 'the dog' here because normally, if you are pointing out a dog resting on a sofa, you are talking about a specific dog in a specific room.