The first is incorrect. This apple is two times bigger than that one.(ok) This apple is twice as big as that one. (Better)

Both are correct! For stylistic reasons, "twice bigger" does not sound very good to us. You can say it - I wouldn't say it sounds bad - it just doesn't sound good. I cannot explain why, but the reason is not grammar. The grammatical structures of "twice bigger" and "much bigger" are identical but for some unknown reason, the second sounds good to a native speaker but the first sounds less good. For the same reason, "more bigger" sounds VERY bad to a native speaker (preferring just "bigger") even though it too is grammatically correct. Another acceptable choice is "bigger by twice": "This apple is bigger by twice than that one" This sort of issue makes the learning of any language difficult. It is always nicer when you can rely on rules to tell you what to do. But sometimes you just have to accept that it is customary to say certain things in certain ways.

Thank you for helping me.