There are lots of cars on the road. There are lots of cars on the roads. Both are correct, with possibly a slight difference in meaning. The more common, and general, way of saying this would be on the road. Road (singular) here is used as a generic concept. Just like,"There are many children in school." School is a generic concept. [That is why your answer key said road was the correct answer.] If you use the plural (roads/schools) you are no longer using the generic concept. So "There are many cars on the roads." means your talking about some specific roads (The roads of America, the roads of Baltimore, the roads of Texas). It should be clear from the context which roads you are talking about. The same can be said of "There are many children in schools." Here, you are no longer talking about the generic concept of school, but precise schools that you have in mind. There are many children in schools (all across Canada, In Quebec, in Toronto, etc.). You see the same thing when you say "The leopard is a fast animal." Singular for generic usage. Leopards are fast animals, is no longer a generic usage, though it is clear that you're talking about all leopards still.

In the above question there are more than one car so cars and these cars on the road because road is one(single) so on the road.

'On the roads' is meant literally. As Alan says, it refers to specific roads in a particular place eg There are many unlicensed vehicles on Britain's roads today. 'On the road' is a set adverbial phrase, similar to 'on the way'. It is often used figuratively as in 'He's on the road to disaster' 'This set her on the road to stardom'. As for your gap-fill question, both 'road' and 'roads' are possible.