There are two aspects of the influence of the Renaissance on Shakespeare. The Renaissance as we usually think about it (being mainly Italian and less pronounced in Northern European countries) happened 200 years before Shakespeare was writing his plays. The Renaissance, literally "rebirth," was a time when people interested in the arts and sciences became interested and inspired by all things classical (language, literature, logic and reason, science, etc. from ancient Greece and Rome) and when scholars/writers became unsatisfied with the practical monopoly that the catholic church had on educational pursuits. Petrarch, for example, deliberately modeled his writing on people like Cicero and the lyric poetry of Catullus and Horace. As a result, he is credited with creating the sonnet form that took off during the Renaissance.

Back to Shakespeare, while most scholars agree that Shakespeare's classical education was a bit of a hodge-podge, he wrote several plays based on classical sources (Anthony and Cleopatra and Julius Caesar, for example, but there are several others, too). He also references classical stories and pagan gods and goddesses from sources like Ovid. In addition, Wyatt and Surrey were responsible for translating Petrarch's sonnets into English, but Shakespeare went a step further and took the whole idea of the sonnet and created the "English" form of the sonnet.

In sum, Shakespeare was influenced by the Renaissance in two ways: He was doing what writers and scholars in Italy were doing by reviving and using classical stories in a creative way. He was also inspired by some of the specific products on the Italian Renaissance (e.g. the sonnet form).