As I see it, the most obvious thing is to replace "I expect it" with "Yes,". When someone asks "Do you think...?", you don't need to repeat "I think..." in your answer". Secondly, "trying to" is redundant since it really doesn't add anything. I say "Yes, because people will be driving to the beach as it is [will be] hot" is shorter and clearer.

I'd rather say ' I think so , ... ' because in the question A 'think' was used as a verb ,'expect ' is not wrong at all though. As for the ellipsis it will be omitting the unnecessary in this case redundant " the traffic will be bad on Saturday". The second sentence ,although not wrong , doesn't sound nice. 'people will be trying to drive ' is not the adequate expression to use here. You want to mention that the streets will be filled with cars striving on the busy road to the beach ( busy due to the hot weather driving people to seek refuge on the beaches ). It could be formulated in a different way such as : I think so, because the heat wave will drive a crowd of cars towards the beach .

Your sentence A is fine, but it could be rewritten with a word that is implied here but not present. Can you see it, and would that be the ellipsis you are looking for? Your sentence B is ungrammatical as it stands. Perhaps it should be something like "I expect so, because people will be trying to drive to the beach, as it's [or perhaps "will be" would be better in order to keep the tenses consistent] hot".