First of all, there was no "Pill." Without artificial birth control, families could not as easily control the number of their children. Secondly, since many children died at birth or during childhood ( many people died during the Tudor period because of the Plague), families produced more children in order to ensure that they would have children that could become their descendents and be there for them in their old age. Thirdly, agrarian families--of which there were many in the Tudor Period--and some others also needed the manual help that children would provide.
During the Tudor period, many people were wealthy landowners and required workers on this land. The lower classes provided this labor. Above and beyond this opportunity for work, the lower classes also had the opportunity to move up socially. For instance, with hard work and luck, a husbandman could move up to a yeoman; as a yeoman, he could purchase a coat of arms and, thus, become a gentleman. For the first time in England, one was not doomed to the station of one's birth.
This upward mobility was the reason that Shakespeare's family became large. As William Shakespeare's father moved to becoming bailiff, his commercial prosperity increased, and so did his family. Eventually, he, too, applied for a coat of arms, but was denied, possibly for having been a Catholic.