Why do we use the n+1 rule in proton NMR?

The n+1 rule is used to describe the relationship between peak splitting and the number of protons a particular proton is coupled to - that is, how many protons are attached to the carbons next to the carbon the proton I am looking at is on? In other words, how many protons are three bonds apart from the proton I'm looking at? There is a condition for this: protons on nitrogen or oxygen atoms don't count as they can't couple to other protons. The reason for this goes beyond a-level. 

The n+1 rule comes from the spin property of protons. Spin is an intrinsic property of protons, just like mass. There are two spin states available to a proton, which we call 'up' and 'down'. Now if we take a proton with two adjacent protons, we can look at all the possible combinations of spins that these two coupling protons can have. These are:
up up
up down
down up
down down
Because we are looking at a proton coupled to two other protons (n=2) we expect the peak to split into three (a triplet) because n+1=3. This happens because the DU and UD combinations are magnetically equivalent, and the NMR spectrum is based on magnetic measurements. Remember we have a lot of molecules of the sample in solution, so we get the 1:2:1 ratio of heights of the three lines in the triplet peak because there 25% of the sample has the UU combination, 25% has the DD combination and 50% has the UD/DU combination. This is why the middle peak is twice as high as the peaks either side. This same logic can be applied no matter how many protons are coupled to the proton of interest. If we have three protons for our proton of interest to couple to (for example, our proton is next to a CH3 group), we get:
UUU
UUD UDU DUU         <--- these three spin combinations are magnetically the same
DDU DUD UDD         <--- these three spin combinations are magnetically the same
DDD
Which gives the 1:3:3:1 height ratio we see in a quartet. Here, number of coupling protons is three (n=3) so n+1=4 and we get the quartet.