It means that you experienced a strong, involuntary muscular jerk, involving many muscles.
"To awake with a start" means the same thing as "to be startled."
In my case, if I am snoozing in my recliner and I awake with a start, people say that I "jump out of the chair." That's not quite true, but I certainly do give a big bounce. And I make a muffled yell, like "WHOAAAA???"
Sometimes people also experience a "hypnagogic jerk" as they are falling asleep.