Prepositions before Romance verbs

A student of mine was puzzled by the following Italian structure: cominciare (to start, to begin, to commence) + a (preposition to) + infinitive. Her comment was that she had never seen prepositions before an infinitive and went on to say that in Czech that is impossible. She is right, but I had never noticed it. There's nothing that prevents Romance verbs to be followed by prepositions, but the same does not hold true for other languages. Other Indo-European groups only allow nouns and pronouns to be followed by a preposition. The same thing happens in English. You say I'm dreaming of a new car but you can't say I'm dreaming of buy a new car. You have to turn that verb into a noun first: I'm dreaming of buying a new car. Where did Romance languages acquire that ability? Not from Latin, at least not from Classical Latin. In Latin you had to do something close to English, you had to turn an infinitive into a nominal form: amare - ad amandum, ab amando, etc. Maybe Late Latin had no problem with prepositions plus verbs and that's how that arose in Romance languages.