In its broadest sense, the Renaissance was a reawakening of the spirit after the restrictive and God-focused Medieval period. The Renaissance brought a sense of freedom to all aspects of life in England, including philosophy, art, and writing. Instead of the focus being strictly on God and the Catholic Church, Renaissance artists and thinkers began to contemplate the relationship between man and God. This allowed writers, in particular, to explore themes they had never been able to examine before this rebirth.
New forms of poetry and writing were developed during this period, but it is Shakespeare's writing which inspired the most change and therefore demonstrated the most influence from the Renaissance. Because plays were so popular and so accessible, everyone from the lowest peasant to the highest nobleman was exposed to his work and therefore to his new way of thinking.
Because England's social hierarchy was no longer so rigid (as it was during the Middle Ages), Shakespeare and others were free to allow every character on stage to experience inner conflict and human suffering. Until now, the religious and noble were exempt from such demonstrations of humanity. Kings and commoners now all suffered the pangs of guilt, the complexities of love, and the torments of indecision (think Hamlet).
Another change in the writing of the Renaissance is the depiction of completely developed, multidimensional characters from all stations of life. No longer were some (royal and religious, in particular) exempt from fallibility and flaws. Shakespeare portrayed them as humans rather than as mere representatives of their classes.
Finally, Shakespeare was able, because of the renewed interest in classical literature, to utilize Greek and Roman mythology into his plays. Such works had been repressed by the Catholic Church during the Middle Ages, but these colorful and useful characters and stories could now be used in everyday life because Shakespeare and others could reference them in their writings. (Perhaps you have read a Shakespeare play and noticed all the allusions to these historical and literary figures.)
Other genres of literature as well as art, music, philosophy, and even science flourished during the English Renaissance; however, in the literary world, there is no better example than Shakespeare of the impact this rebirth had on English literature.