Mario's explanation is perfect, I want to add : the infinitive is the form of the verb that act also as noun. The name of one action (llover, correr, etc. ) , is asubstantive. When, in a sentence, the verb ("vi") needs Direct Object, this DO can be: -a noun (Esta tarde vi lluvia) -an infinitive (Esta tarde vi llover) -a sentence working like noun (Esta tarde vi que llovía)

Though this is a song and uses poetic style, this structure is very common. I will give you a few examples. "Quiero tanto a mi perro porque lo único que él hace es quererme" "I love my dog so much because the only thing he does is to love me". "Veo a gente leer todo el tiempo en el metro" "I see people reading all the time on the metro". As for "Esta tarde he visto la lluvia. He visto gente corrian", should be "he visto gente que corría". "Gente" though it refers to a group of people, thus plural, it is a singular noun, so the conjugation cannot be "corrían" because it is plural, you have to use the singular "corría". The plural of "gente" is "gentes", rarely used. I hope this helps you.

Spanish is much more "Poetic" in style than English so many things cannot translate literally. (Ví llover) is much more common in a song than in the actual spoken language. Also this style of speaking seems much more common in music from the south of Spain and Mexico and in some salsa music. The spoken language can differ from the formal written language, so keep this in mind when listening to music in Spanish.

Hi, I am a native speaker of Spanish from Mexico (as was Armando Manzanero, the writer of the song you refer to, by the way). Although I agree with the other comments about the sentences "vi llover, vi gente correr" being "poetic", I have more to say about that. The sentence "vi llover" would be perfectly normal in non-poetic, ordinary speech, used for example when you just look out of the window while it rains. The sentence "vi la lluvia" would mean (to me) that you actually SEE THE RAINWATER FALLING (and perhaps nothing else matters to you), while "vi llover" suggests (to me) that you SEE WHAT HAPPENS WHILE IT RAINS. On the other hand, the sentence "vi gente correr" is less natural (though perfectly grammatical), and the author just used that verb form to make it rhyme with the previous verse. For me, a more natural sentence to express the same would be "vi gente corriendo". Good luck ;)