How do "proofs" work in mathematics, down to the basic level? I came across this basic example of a problem:

Prove by mathematical induction that 1 + 2 + 4 + 8 + 6

What am I supposed to prove, the sum? I don't quite get it, so I decided to dig in to encyclopedia references for help:

In mathematics, a proof is a deductive argument for a mathematical statement.

So a proof is just arguing that the "problem" to the equation is correct? Under what measures confines something as an argument of proof? Anyone can argue anything.
I ended up in axioms:

An axiom, or postulate, is a premise or starting point of reasoning.

So if I "reason and/or argue that my problem is correct" I am "proofing"?
Is that all that means?
So 1 + 2 + 4 + 8 + 6 = 21. Assuming you have elementary addition knowledge, and can add summands together, where is there an argument and why?
Basically, what is the point of "proof" in this situation, and how does the idea of "proofs" in math make its purpose worthwhile?
I mean I have a well open mind, but you can't realistically think everything has a purpose for everything at all times. I don't see any practical understanding or idea on "proofs" that makes me just "get it" and find it useful.
 A: A proof in mathematics is the same thing as a proof in every day life: it's an argument that convinces you that something is true. The only difference is that in mathematics we have much stricter standards for how convincing the proof should be.
Now then. Your initial confusion was about what this means:

Prove by mathematical induction that $1 + 2 + 4 + 8 + 6$.

You were right to be confused, because that is a completely meaningless sentence. MJD's comment underneath your question does a good job at explaining the problem. I think we'd all appreciate if you could post exactly where you found that sentence, because I suspect either you misread or misunderstood it, or it was written by an idiot.
Now, if you were asked prove that $1 + 2 + 4 + 8 + 6 = 21$, that's a different story (again, see MJD's comment if you don't understand the difference). As you said, you can prove that just by doing basic addition, and no reasonable person (even a mathematician) would ever ask for any further proof than that. Again, I suspect you must have misunderstood the original question (please post a link or a reference!).
You ask:

...if I "reason and/or argue that my problem is correct" I am "proofing"?

This sounds basically right, although your use of the word "problem" is strange. How can a problem be correct? A statement can be correct. You prove a statement. Like I said in the first paragraph, "proof" doesn't have any mysterious technical meaning in mathematics (except in super-advanced, Phd level logic classes), it's just that mathematicians are quite a bit stricter about how certain they want proofs to be.
Now, I don't quite understand what you mean when you say "you can't realistically think everything has a purpose for everything at all times", but what you seem to be asking is what the purpose of proofs is in mathematics. There are a few purposes:


*

*The simplest reason is that although $1 + 2 + 4 + 8 + 6 = 21$ is obvious, there are a lot of things in math that are far less obvious (but still true). If I just told you that the area of a circle is always equal to its perimeter times half the radius ($A=P\times\frac{r}{2}$), there's no reason why you ought to believe me. It doesn't seem all that obvious. But it's true, and you can prove it to be true.

*Probably not so much at your level, but as you progress to higher mathematics, you more and more often come across things that seem true, but are actually false. So you start to trust your instinct a lot less, and proving things carefully becomes much more important.

*A simple fact like $2+2=4$ is pretty obvious and not all that mysterious, but in higher mathematics many things just make you ask why?. Why is this fact true? A proof also acts as an explanation as to why something is true.

A: It seems that what you are after are the means of proof. These are studied in the subfield of logic.
The argument in the proof of a statement like 2+3 = 5 resides in several steps, which at first are not at all that obvious.
First, the symbols 2 and 3 denote numbers and the symbol + defines an operation of these numbers that have precise mathematical definitions and/or axioms that describe their function. 
For instance, in the axioms defining the (natural) numbers, we postulate the existence of a "first" element, and we give it a name/symbol, say "1" - axiom N1. Then we postulate that all numbers have a successor, the successor of x is written as S(x) - axiom N2. Note: we did not say what this successor function does yet, but we can derive the existence of S1, S(S1), S(S(S1)), etc. We would like them to be different. So, we postulate "Sx = Sy implies x = y" - axiom N3. 
Now, we can prove that if any such successors are equal, then we have only one element. E.g. if S(S(1)) = S(1), then S(1) = 1 (using axiom II); hence S(S(1)) = S(1) = 1. Repeating this argument for any successor of 1, completes the proof for any such statement. Certain proof-techniques give us the option to prove the general statement. (It's called Induction).
We can add a fourth axiom which prevents this: $x \neq Sx$ - axiom IV. etc.
From these, we see that we can identify the successors of 1 with the numbers 2, 3, etc. of our normal counting experience. We have not yet defined addition. We can define addition with other axioms. E.g.
A1: x + 1 := Sx
A2: If v = Su, then x + v := S(x+u), 
etc.
From there, we can prove that 2 + 3 = 5 by realizing that 2 = S(1) and 3 = S(S(1)), so 
2 + 3 = S(1) + S(S(1)) = S(S(1)+S(1)) = S(S(S(1) + 1)) = S(S(S(S(1))) = 5, where we use A2 with different values of x, u and v, etc.
