The idiom does mean excited, but only in one specific context. Imagine you are in a cinema or theatre. If the film or play is boring, you'll be sitting far back in your seat, falling asleep. But if it's exciting and gripping or scary, you'll be so keen to find out what happens next that you'll be leaning right forward, determined not to miss a moment of the action. You'll be on the edge of your seat. 'The plot of the film was so clever that it everyone on the edge of their seats from beginning to end.'

You can use the idiom more widely than just when at a cinema or a theatre. It means you are excited, or stimulated, but you are generally in a passive state, rather than running around the place. You could be at the edge of your seat at a football match, driving, being a passenger in a plane, at a meeting etc. "It was real edge of the seat stuff, I didn't know what he was going to come out with next" To come out with meaning to say something that would shock or embarrass, and to do so in an unpredictable manner.