Why does non-continuous sleep makes you feel tired?
Sleep is not homogenous - there are different phases of sleep - REM, NREM-1 to NREM-4. And these different stages may play different roles in 'rejuvenating' the body (as you put it).

When you first fall asleep, you go into NREM-1 phase, which is the lightest sleep, and progresses into NREM-4, and for the first half of your sleep cycle, this cycles so you go from NREM-1 > NREM-2 > NREM-3 > NREM-4 > NREM-3 > NREM-2 > ....

On the first half of your sleep cycle, the NREM3 and NREM4 phases dominates, only on latter half of your sleep cycle are REM phase observed and progressively dominate.

Depending on how much you are waken up by, once you escape your sleep cycle, you may re-enter back at a different stage, which may mean you are no longer able to have much REM sleep. Subsequently, you will not have the rejuvenation that is associated with REM sleep and you feel tired.

Interesting note: There are people who have tried a method called polyphasic sleep, where they nap/sleep periodically and regularly, instead of sleeping once a day for 8 hours, say. People who promote this says that people do not feel tired because as soon as they nap, they go in REM sleep, and so the amount of REM sleep you get per day will still be the same. I'll let you make your own mind up about this one.