Learn Spanish (Latin America): Me molesta que el humo entre a la casa. - It bothers me that the smoke enters the house.

Because the sentence starts with Me molesta que (It bothers me that). In Spanish, whenever you express an emotion, judgment, or feeling about an action performed by someone or something else, that action must be in the subjunctive mood. That is why the regular verb entrar becomes entre.

Why do we need the preposition a in entre a la casa? In English, we just say 'enters the house.'

Unlike English, the Spanish verb entrar cannot attach directly to a place. It usually requires a preposition to show movement into that destination. In Latin America, entrar a is the most common and standard way to say 'to enter [a place].'

Is Me the subject of the verb molesta?

No, Me is an indirect object pronoun meaning 'to me'. The verb molestar works exactly like gustar. The actual subject of the sentence is the whole idea of the smoke entering the house. Since that entire idea counts as a single concept, molesta is conjugated in the singular.

Could I just use the infinitive and say Me molesta el humo entrar a la casa?