The divisions between 0 and 1 show how a fraction is made up.
Let's take the case where the interval is divided up into 12 equal divisions. One division represents 1/12.
Two divisions represent 2/12 which together represent 1/6 because 6*2=12 and the interval between 0 and 1 contains 6 sixths.
Three divisions represent 3/12 which represent 1/4 because 4*3=12 and the interval between 0 and 1 contains 4 fourths or quarters, just like there are four quarters to $1.
Four divisions represent 4/12 or one third.
Five divisions just represent 5/12 and 7 divisions represent 7/12; but 6 divisions represent 6/12 or one half and two lots of these make 1. So 2*6/12 is the same as 2*1/2. 11 divisions make 11/12.
Eight divisions is 8/12. Since 4/12 is a third, 8/12 must be two thirds because 2*4=8 or 2*4/12=8/12.
Nine divisions are 3*3/12=9/12, and since 3/12 is the same as a quarter, 9/12 must be three quarters.
Ten divisions make 10/12, but since 5*2/12=10/12 and 2/12 is one sixth, 10/12 must be 5 sixths or 5/6.
If the interval 0 to 1 is divided into 60 divisions we get more fractions. The divisions help us to add and subtract. The number of divisions is associated with the least common multiple (LCM). So the LCM of 2, 3, 4 and 6 is 12. For 60 as LCM we have 2, 3, 4, 5, 6, 10, 12, 15, 20, 30. These numbers of divisions give us the fractions 1/30, 1/20, 1/15, 1/12, 1/10, 1/6, 1/5, 1/4, 1/3, 1/2. So to add 1/5 and 1/6 we use the divisions 12/60+10/60=22/60=11/30 because 22=2*11 and 22/60=2*11/60=11/30. We can also see that 11/30-1/5 is the same as 22/60-12/60=10/60=1/6.
This should give you some idea how dividing 0 to 1 into different divisions helps you see how fractions add and subtract.