How to understand why $x^0 = 1$, where $x$ is any real number? Alright, so the idea of an exponent, $x$, is that you are multiplying its base by itself $x$ number of times. With base $5$ and $x=3$, we have that $5^3$ = $5 \cdot 5 \cdot 5$

I understand that the logarithm with base $a$ of $x = c$,

tells us that

$$a^c = x$$

and for $c =$ positive; values for $x$ are greater than $1$, and for $c =$ negative; values for $x$ are less than $1$,  and for $c = 0$, values for $x$ are...$1$.

So in short, I understand how, by means of observation of the graph of $f(x) = \log x$, we can see that $f(1) = 0$, BUT, I see no other way to understand why $x^0 = 1$, apart from the graph and everything around that very point.

I honestly cannot get my head around the idea, "$5$ times itself $0$ times... is one".

Is it that there is no fundamental answer for this but that we simply know by the graph? Or can I truly understand $x^0 = 1$ on its own?
$x^{n+1}=x\cdot x^n$ right?

so 

$x^1=x \cdot x^0$ but $x=x^1$ so for that to hold true, $x^0$ must be $1$.

Similarily,

$\large x^{-n} = \frac{1}{x^n}$.

So $\large x^n \cdot x^{-n} = x^n \frac{1}{x^n} = 1$. 

But $\large x^n \cdot x^{-n} = x^{n+(-n)} = x^0$, so once more, $x^0=1$.

There are really many reasons for that to hold, and all of them are just a consequence of some agreements we've made previously.
First of all you(not only you many) have to understand mathematics is also a language, another way of communication. Language is obviously based on grammar( lexical rules). Power is a operation which is based on multiplication. $2^2$ means its just a language which convey you that you have to multiply 2 number 2 times. $2^0$ means you should not use 2 at all. Automatically our mind fills the number 1 instead of void.