Why is random noise assumed to be normally distributed?
Starting at an even more basic level than the other (much smarter) answers, I'd like to pick up on this part of the question:


  This seems contradictory to me as on one side it is random then on the other side their distribution is considered normally distributed.


Perhaps the issue here is what ‘random’ means?

To be clear: ‘random’ and ‘normally-distributed’ do not contradict each other.  ‘Random’ simply means that we can't predict exactly what the next value will be.  But that doesn't mean we can't make probabilistic statements about it.

Consider two experiments:


If you throw a (fair) die, then it could show any number from 1 to 6.  We can't tell which number will come up, but we can say that all numbers are equally likely (i.e. the distribution is uniform).
If you throw two dice and take their sum, that can be any number from 2 to 12.  Again, the sum is still random — we can't predict what it will be — but we can say that those values are not equally likely.  (For example, 7 is six times more likely than 12.)  So in this case it has a non-uniform distribution.  (You can plot all the probabilities; they take on a peaked shape a bit like a normal distribution.)


So there's no contradiction: both cases are random and have a known distribution.

In fact, most things that are random tend to have a non-uniform distribution: electrical noise, weather, the wait for the next bus, voting patterns…  Being able to make general statements about them without being able to predict the exact values is one of the strengths of statistics.

(As for why you often end up with a normal distribution, that's a result of the Central Limit Theorem, which says that when you combine many independent random variables, the result tends towards a Gaussian (normal) distribution.  So you see that crop up a lot.)