Everybody knows the Renaissance meant a return to classical Greek and Latin texts. But where did these texts come from? How did they get to e.g. Padua, Paris or Cambridge? And was the language they were written in the only reason to suppose they were more authentic than the Arabic translations?

Classical texts arrived in Europe from different places at different times. From around 1300 onwards, scholars scoured the monastic libraries of Europe to collect them. The main sources of classical texts were:

The Carolingians (courtesy of Irish missionaries, who brought these works into continental Europe in the 7-8th centuries). The scholars of the court of Charlemagne (742-814) and his successors formed a loosely-organized movement later historians have dubbed the “Carolingian Renaissance.” Among its other projects was the collecting and copying of the texts of Latin antiquity. In fact, around 90% of our surviving classical Latin texts were copied and preserved in monastic libraries by Carolingian scholars. Part of Charlemagne’s scholarly program involved the reform of handwriting. The result was a new written script later called “Carolingian minuscule,” which is essentially the origin of the letter shapes we still use today. With no way of yet created to critically or scientifically date texts, when Renaissance book-hunters found Carolingian manuscripts, they initially believed that the clear writing of Carolingian minuscule—much more legible than their own Gothic script—must be the writing of ancient Romans. They therefore called it “littera antiqua.”
A second wave of ancient manuscripts entered Europe during the long twelfth century when universities were just forming and scholars were traveling to Spain (Cordoba, Toledo, etc.), Sicily, and North Africa to hunt for classical texts translated from Greek and Latin into Arabic and then into Hebrew and Latin. This is when the complete (mostly) works of the Greek philosopher Aristotle reached the west and had a profound effect on western thinking, especially through the application of Aristotle’s works on logic and science. [Gerard of Cremona] (http://en.wikipedia.org/wiki/Gerard_of_cremona) (d. 1187) is an interesting figure in this regard; he translated dozens of Arabic works into Latin.
A third wave of Greek texts reached the west in dribs and drabs from Sicily and the Byzantine Empire and then in much greater quantities in the 15th century, especially after Constantinople fell to the Turks in 1453. Remember that the Renaissance began as an effort to recover Latin writings; very few scholars in the Middle Ages and early Renaissance knew Greek. And Greek remained a specialized language even during the Renaissance. All scholars spoke and read Latin since they were educated in Latin; Greek had to be specially learned. The first great teacher of Greek to the Latin west was [Manuel Chrysoloras] (http://en.wikipedia.org/wiki/Manuel_Chrysoloras), who was invited to Florence in 1396 specifically to teach Greek. The philosopher largely responsible for reintroducing Plato to the west, was [Marsilio Ficino] (http://en.wikipedia.org/wiki/Ficino), who translated his works into Latin and published them in 1484. (This is an overview. I’ll let folks better informed on Greek translations refine what I’ve written.)

EDIT: I forgot part of your question. The Greek texts the Arabs translated were themselves not always accurate and were further removed from the original when put into Latin. When Renaissance scholars found the original Greek versions of works that had passed through Arabic into Hebrew or Latin, they could see that what they had had earlier was not as accurate as the original.

Sources: Though it should be used with reservations, a decent summary of Renaissance collecting of classical texts is Stephen Greenblatt’s The Swerve: How the World Became Modern (2011). A more hands on texts (and much used by Greenblatt) is Phyllis Gordon’s Two Renaissance Book Hunters: The Letters of Poggius Bracciolini to Nicolaus de Niccolis; Translated from the Latin and Annotated, which shows book-hunters at work finding texts.