Why is the numbering of computable functions significant? My course is about computability theory, and I'm having troubles with one of the main concepts. This might be a really newb question, but I've been struggling with understanding it's significance (and I was stupid enough to not ask my teacher while in class). 

Can someone give me an example in terms of something practical to why numbering 
computable functions is helpful?

I'm a computer science student, so maybe an example from that field might help me understand.

EDIT: Okay, so with the assumption then that the halting problem is the main reason we number computable functions. 

In that case, numbering the functions should help us always prove if a function will halt or not. So, say we take the gödel numbered program $P_{2057}$, which stands for the program S(1), S(1), J(2,1,1). This represents the function x+2, over the domain of $\mathbb{N}$. 

How does knowing the number 2057 help us decide if this program will halt or not?

EDIT2: I tried reading the papers that @sav linked to me, although I find them relatively difficult to understand, as I'm not that good with theoretical things yet. 

Our course book is Computability, An introduction to recursive function theory by Nigel Cutland. I've found online many people praising this book for being clear and easy to understand, but for me that hasn't always been the case. The teachers explanations don't always make sense either, and sometimes it feels like the book is the teacher, not her(which is one of the big reasons I finally decided to come here for help).

Therefore, another question I have is, the book claims the numbering of functions to be of utmost importance in both the s-m-n theorem, and universal functions. After reading through those parts multiple times, I still don't understand their significance, and yet they are constantly referenced in later parts. Could someone maybe try and explain this connection in another way?
You have already been numbering the computable functions, you just didn't know it at first. The first time you saw a definition of the computable functions, you saw it in the context of some specific model of computation, such as register machines. Each machine can be viewed as a "number" for the function it computes; by standard coding methods we really can represent each machine by a natural number, justifying the terminology. (For example, as soon as you can type something into a text file, you can view the file as a sequence of 0s and 1s. Prepending a 1 gives a single number representing the data you typed).

So a number is nothing more or less than a program for computing a function. Rather than viewing the program as a sequence of complex symbols, we can view it as a sequence of 0s and 1s, as a number. Of course, to make use of the program, you have to know what model of computation it was written in. 

There are many different models of computation: register machines, Turing machines, WHILE programs, $\lambda$ calculus, $\mu$-recursive functions, etc.  Each of these, when fully specified, gives a different numbering of the set of computable functions.

A key point in computability theory, which is not always obvious in a first course, is that the main focus is not on any of these specific models of computation - the focus is on the computable functions themselves. 

So, rather than attaching ourselves to any specific model of computation, we just assume that we have a reasonable numbering of the computable functions, and work with that numbering abstractly. "Reasonable" here means that the numbering has properties stated by the  s-m-n theorem, the recursion theorem, and the universal function theorem.

So, when you ask Why is numbering the computable functions helpful?, I would respond that they are already numbered! Why focus on "numbers" abstractly? We could just as well call the numbers "programs", and call the numbering a "programming system". Nothing new would come from that terminology change. 

It is true that, if we have a numbering of the computable functions, knowing that some function has number 2234 instead of 2332 doesn't really mean much. The somewhat surprising thing is that the same is true in all the standard numbering systems. The undecidability of the halting problem shows, for example, that just knowing a particular register machine to compute a function is not sufficient, in general, to let you determine whether the function halts on given inputs. 

In computer science, they would call this a "static analysis" of the program. The undecidability of the halting problem shows that static analysis has significant theoretical limitations.  But this is true in every reasonable programming system, and we know that because the theorem about the halting problem is proved abstractly starting with an abstract numbering of the computable functions.  We can prove it once and for all, rather than proving it separately for every model of computation.
The numbering is just a means to an end, and is significant only in the context of the proof in which it is used.

Consider an analogous example. Suppose we have 3 pigeons and 2 pigeon-holes. If each hole can hold only 1 pigeon, prove that there will always be at least 1 pigeon not in a hole. One answer: number the pigeons 1,2,3 by the order in which they enter a hole (exclude simultaneous entry for simplicity). Then by the time pigeon 2 enters, no holes are left for pigeon 3.

Numbering the pigeons just helps with this proof, with no claim to significance of the numbering outside the proof. Numbering the computable functions is similar.