How was Elizabethan theater important to people that lived during the time of Shakespeare? How did it affect the lives of Elizabethan people?
I think the first thing to remember is that plays were, for the most part, not high-class culture back in Elizabethan times the way they are today. Today, we think of Shakespeare as really high class, but it wasn't that way back then.
Instead, theaters were pretty much entertainment for everyone. This was before TV and movies and such, of course, so they turned to other things for entertainment. Theater was a form of entertainment, along with things like bear fights and dogs fighting rats and other such things.
The rich did go to theaters, but so did the poor -- they were the "groundlings" who stood in front of the stage in poor conditions to watch. You can see in Shakespeare's plays how he caters to these people with coarse jokes and things like that.
So the theater was really just another form of entertainment in those days.
Shakespeare's plays may be thought of as editorial commentaries of the day. For example, Henry V was important to the current times when Shakespeare wrote it because the people could see a correlation between Henry and Elizabeth I. Henry's lineage led to Elizabeth confirming her right to the throne. His mastery of the realm and war and the people reflected on the mastery of Elizabeth for her realm. Henry V could be said to have been written in homage to her.