Shakespeare's plays generally lacked anything in the way of sets at all. There might be props which would be used to create sound effects, so that the audience could imagine that they were privy to a thunderstorm, for example, but the stage would generally be free of all but the most basic "furniture," and the audience would be quite used to imagining what was supposed to be going on behind the actors.
Note that in modern stagings of Shakespeare's plays, the curtain is not used to signal that a scene is over. The curtain will usually be pulled only when there is an interval in the proceedings. At the end of a scene in a modern production, however, we might see the lights dimmed, a feature to which Shakespeare's early plays did not have recourse. In some sparser modern productions, though, scene changes are still signaled in much the same basic way as they were in Shakespeare's day: when all the characters in a scene have left the stage—exeunt—then the scene is over. In his language, too, Shakespeare gives us an indication that a particular scene has been brought to a close, in that the final two lines of a scene are very frequently a rhyming couplet. This concluding rhyme gives a signal to the audience that this particular conversation, or incident, is over.