Gringuita is the diminutive form of 'gringa', a colloquial term often used in Latin America for a foreign woman, especially one from the United States. The '-ita' ending adds a playful, affectionate touch, sometimes with a teasing edge.
In the song, "Llégale, llégale, mira que se ponen gringuita'" ("Pull up, pull up, look how they turn all gringa") describes local women adopting a more 'foreign' look or attitude after spending time around American tourists in Miami.
A Venezuelan band and a Colombian singer built a whole song out of a joke every Latin American family knows: you send the kid to Miami for six months of English class, and English is not what comes back. "¿Por qué tiene que ser así? Seis meses aquí" (Why's it gotta be like this? Six months here), and then the punchline, "Ay, mami, nadie aprende inglés en Miami" (Oh, babe, nobody learns English in Miami). What she did learn gets listed with no embarrassment at all: nights at Kiki and La Otra, Tito's vodka, the beach, and "No aprendiste inglés, pero aprendiste otra cosa" (You didn't learn English, but you learned something else).
The best moment is the one the song takes back a second later. It reaches "Donde yo te abriría y después te cerraría" (Where I'd open you up and then close you), lands on "Las piernas en el balcón, digo" (Your legs on the balcony, I mean), and hurriedly corrects itself to "Puertas del balcón" (Balcony doors). Then the joke runs out. He flies home "pasado mañana" (the day after tomorrow), his two languages have folded into each other, and the only sentence left is "te voy a miss you" (I'm gonna miss you). Six months of classes for one phrase.