Translation:You cannot smoke in this restaurant.
Multiple はs aren't necessarily wrong, though they're not common either. The literal meaning of why は is in this form is because the verb is the topic for いけません which means "will be not good."
The て form has many uses. One of them is to allow multiple verbs in a single sentence (Example: "I ate fish and drank water" translates to さかなを食べて水をのみました). Normally, only the final verb is regularly conjugated, which is why すって is used. The literal translation of this sentence is "To smoke in this restaurant will not be good," which is translated to English as "you cannot" or "you must not."
Bear in mind that this explanation was never given to me and I am inferring it based on all that I have learned so far.
It marked "You may not smoke inside this restaurant." as incorrect for what it claimed was two reasons. First, it wanted "in" instead of "inside." I am pretty sure that this is a distinction without a difference. [I don't think you can be in the restaurant without being inside of it. ] Second, it said it wanted "cannot" instead of "may not." But this is a permission thing. About the only reason you "cannot" smoke is if there is no oxygen inside! I would argue that in English, not only should it be "may," but that "can" (in all reasonable likelihood) is incorrect.