Why do they call it base 10? Now, I know intuitively why it's called base 10: because there's 10 numbers.
But see here's the thing, if we're working with numbers 0-9 (and of course we are), we use up our numerical artillery at 9. Why isn't it called base 9?
If it's because 10 is the number we round up to,that logic doesn't seem to transcend to other bases because every base would literally be called base 10 if we were to extend that logic.
So, in using a system with 9 unique characters, and calling it base 10, this leads to inconsistencies with every other single base. Take binary, base 2, it's called base 2 though uses 1 character, hexadecimal, called base 16, uses 15 char (1-9 [not 10 though] and A-E).
Now in binary, octal, and hexadecimal, it's not really an issue, but what about an arbitrary base n? What do we call it then?
We could assign a name to it like we have with 'base 2' i.e binary, but if we want to work within a new base, then to communicate you're working in base n, it becomes tedious. 
The most logical thing to do, in my opinion, would be to call the base by the last unique number you have to work with. This is easier to remember, and resolves the issue of nomenclature.
 A: The name refers to the way we choose to group our items. The way we've evolved, we found that nine counting symbols and a symbol for nothing suffice for our preferred base. We're able to recycle the glyphs $1$ and $0$ to denote our grouping, which is given by the combined symbol $10$.

To illustrate why it's all about the grouping, and not about the final unique symbol, let's roll with this example. Pick an amount of symbols that can use to enumerate your fingers. You may prefer $$1,2,3,4,5,6,7,8,9,X$$ but another could have just as easily chosen $\odot,\triangle,\square,\cup,\star,\vee,\uplus,\times,+,\LaTeX$. These are just glyphs at this point, markers for each of the fingers. For convenience, let's use my suggested set. 
We have an issue, I want to add more symbols to keep track of toes. Okay, how about:
$$1,2,3,4,5,6,7,8,9,X$$
$$1',2',3',4',5',6',7',8',9',X'$$
Man, that's a lot of symbols. I like the number of my fingers, $X$, and in fact, you could say I have $X$ toes. Wow, grouping things by $X$ is convenient, both physically and visually. Can I keep track of these groups by repeating my symbols to enumerate larger quantities in groups of $X$? 
Well, it would be nice if I started counting from nothing, so let's add a symbol for that.
$$0$$
$$1,2,3,4,5,6,7,8,9,X$$
Now, how should this work? Well, I can have $X$ objects. Or I can have $X$ groups of $X$ objects each. Or, wow, even $X$ groups of $X$ groups of $X$ objects! It seems like reaching $X$ marks a sort of restart point. I could denote this regrouping by saying I have $1$ $X$ and $0$ other stuff. Okay, I feel inspired. Let's combine those symbols to replace $X$:
$$0$$
$$1,2,3,4,5,6,7,8,9,10$$
And on and on it goes... This was more or less a colorful way of expressing BigBearzzz's answer: each digit's location denotes a higher grouping. We don't group every nine objects, though. We group ten objects.

What about other bases? Well, say we choose to group every three objects. We would still want a symbol for nothing, $0$, another for one $1$, for two $2$, and once again we can recycle our glyphs to call one group of three $10$. That's where we regroup. Note that these are just glyphs, recycled from the base ten representations. The meaning is different here than in base ten due to where I choose to regroup. Why is this called base three? Well, I'd like to refer to this new system, and I choose to borrow the name from a convenient source: my favorite base of all, ten.
A: When I work with numbers in base 16 or base 2, the string "10" is not called "ten", but "one zero". We generally don't have names for specific integers that reflect some other base, so although a full 16 bit value is ffff and then 1 0000 is significant to me, I don't have a spoken name for it that corresponds to thousand.  It's only known as "sixty four K", which is nothing like how it's written.
The spoken "sixteen" refers to the specific natural number, no matter how else it is noted. The English name comes from a base-10 herritage, but we could have had a special word for it that stands alone, like dozen, gross, or mole.  Maybe someone will introduce one and it will catch on.
A: This is more like a comment but it's too long, so I'm putting it as an answer instead, please accept my apology. 
In base $10$, the "symbol" $78152_{10}$ represents the number 
$78152_{10}=7\cdot 10^4 + 8\cdot 10^3 + 1\cdot 10^2 + 5\cdot 10^1 + 2\cdot 10^0$.
In base $n$, the "symbol" $78152_n$ represents the number
$78152_n=7\cdot n^4 + 8\cdot n^3 + 1\cdot n^2 + 5\cdot n^1 + 2\cdot n^0$.
You can see that we prefer to call a base by the number that is raised to the power of its position from the last digit. That is why we call it the base.
A: You're mixing the actual number with its representations. Yes, in base $n$, the string $10$ represents exactly $n$.
But now forego of the decimal representation of $10$, and think about it as "how many digits a healthy human being has on both their hands". This is your $n$, now. Let's for the sake of simplicity call this number "ten".
Now we are counting in base ten, and that's that. We are taught to innately think in terms of this base, which is why we call it base $10$.
