HOw did it work if he was not body jumping?
Once Sam stepped into the quantum accelerator thingy, he never stopped jumping from body to body. He never went back "home".
There in his own body with an illusion that made him look and sound like the person he was displacing to others? Was the illusion light & sound or mind control?
Never explained. In the beginning of the series, I think it was actually that only his mind was "leaping", but later on they wrote it as being his whole body - body-switching with people in the past. Which annoyed me because I think they did it for the amusement value of having big tall Sam walking around in dresses and high heels - with no explanation of how he could fit into them, or into the clothes of a man smaller than him, either. It just made no damn sense at all, and the show was stretching disbelief pretty hard anyway.
What happened to the person he replaced while he was there?
They had an alien/government abduction experience. They woke up in the Waiting Room, in Sam's body, and probably ended up believing it was all a hallucination or a dream. I always wondered what happened to them after they went back to their own bodies, after Sam fixed their problems. They'd have no recollection of what Sam did while he was "standing in for them", and he always made major changes in their lives - for the better, of course, that was the premise, but still - if you wake up from an intensely vivid and weird dream to find that days have passed, you've apparently done things you cannot recall, and your life is different and you've no idea how it got that way - isn't that, itself, kinda going to mess up your life? Sam was basically putting all those people in danger of being committed, of being thought insane or thinking themselves insane, and disappointing and/or frightening all the people around them that he'd managed to save/woo/impress/whatever?
And yeah, the series finale sucked. It was cool, right up to the point where we realized it made more questions than it answered, and we never really found out what *really* happened to Sam.
Wow, I have strong feelings about Quantum Leap. I watched that show during a very difficult part of my life, and it gave me something to enjoy and lose myself in, but I've never been able to re-watch any of the episodes with the same enjoyment.