They had quite a substantial influence, dating all the way back to the time of Petrarch and Boccaccio. They were the first two westerners who made serious attempts to read Greek literature since antiquity, and though ultimately they failed, they made some important first steps. They used two teachers, both Calabrian Greeks with ties to Constantinople, named Barlaam and Leontius Pilatus. They were primarily interested in learning Homer, which was unfortunately a very difficult text for beginners, and so neither Petrarch nor Boccaccio ever really learned Greek (though Boccaccio was somewhat successful. You can see many excerpts from Homer in his Genealogy of the Pagan Gods).

A generation later, a man by the name of Manuel Chrysoloras came to Florence from Constantinople, and it is with him that the Greek influence really takes off. He chose simpler texts for instruction, and he also managed to simplify the grammatical rules of ancient Greek substantially. From then on, knowledge of Greek was an important element of a humanist education.

Many Byzantine refugees were also responsible for bringing Greek texts over to Italy from Constantinople. We have frequent mentions of scholars hiring people to buy books for them from the East. Cardinal Bessarion, for example, was a Greek refugee who brought a huge Greek library with him to Rome.

If you would like to read more about any of this, I have a few books to suggest. For a very general and brief overview, I suggest Colin Wells Sailing from Byzantium. For some more heavy scholarship, you will unfortunately need to turn to Italian sources (unless you want to read about Bessarion, but unfortunately I don't have any specific reference for you on him). Petrarca e il mondo greco I: Atti del Convegno internazionale di studi Reggio Calabria 26-30 novembre 2001 and Manuele Crisolora e il Ritorno del Greco in Occidente: Atti del Convegno Internazionale (Napoli, 26-29 giugno 1997) both have a number of interesting articles in them. In English, N.G. Wilson's From Byzantium to Italy: Greek Studies in the Italian Renaissance is quite good, but also very dry.

I hope that this helps, let me know if you have other questions!