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.

In this context, it means you suddenly woke up, feeling alarmed or surprised. For example, you were sound asleep when you felt someone stroking your heel. You knew you were alone in the house, so you woke [up] with a start.

很高兴! It sounds '惊觉, 吓醒' in Chinese. In this phrase, 'start' means startle, surprise, astonishment, and alarm. e.g.) I jumped with a start when hearing gunfire. She came to with a start. He raised his eyes with a little start.

abruptly, suddenly, quickly (usually) from surprise or fear