The task i need explaining is: how to show that an estimate of the mean time spent on homework is 64.8 minutes.

It's just an estimate.  There's no telling how close it is.

To estimate using just the information in the table,
ASSUME that the average of all the students in each
slot is the average time of that slot.

I know that's confusing.  I can't think of a better way to say it,
so here are two examples of what I mean.  Look at the table:

-- 6 students said that they spent between 0 and 30 minutes.
   ASSUME that those 6 students averaged 15 minutes each.

-- 21 students said that they spent between 60 and 90 minutes.
   ASSUME that those 21 students averaged 75 minutes each.

So, when you add up the times for all 50 students, you'll have

 (6 x 15 min) + (14 x 45 min) + (21 x 75min) + (9 x 105 min) =

When you total up all those times, divide it by 50 to estimate
the average per student.  

Remember ... it's only an estimate.
If the first group had 1 student that spent 2 minutes, and
the other 5 of them spent 29 minutes, then it won't work.
But you'll never know.