This isn't as rigorous as the accepted answer, but explores the question a bit from another perspective.
Take your example,
$(a+b)^3=\binom{3}{0}a^3+\binom{3}{1}a^2b+\binom{3}{2}ab^2+\binom{3}{3}b^3$
Let us consider this small rewrite of the same example: $(a+b)^3=\binom{3}{0}{a}\cdot{a}\cdot{a}+ \binom{3}{1}{a}\cdot{a}\cdot{b}+ \binom{3}{2}{a}\cdot{b}\cdot{b}+ \binom{3}{3}{b}\cdot{b}\cdot{b}$
Consider that the binomial if expanded would take on the following product:
$(a+b)^3 = (a+b)(a+b)(a+b)$
To compute this product we have to pick one summand in each of the $(a+b)(a+b)(a+b)$ and multiply them together. We then do this in all possible ways adding each term together. Since there's $2$ possibilities for each of $3$ choices there's $2^3=8$ total possibilities. In essence, we are forming all length 3 strings over the alphabet $\{a,b\}$.
That may be hard to follow, but it's easy to see it written out:
$(a+b)^3 = (a+b)(a+b)(a+b) = $ ${a}\cdot{a}\cdot{a} + {a}\cdot{a}\cdot{b} + {a}\cdot{b}\cdot{a} + {b}\cdot{a}\cdot{a} + {b}\cdot{b}\cdot{a} + {b}\cdot{a}\cdot{b} + {a}\cdot{b}\cdot{b} + {b}\cdot{b}\cdot{b}$
So there's $8$ possibilities, so let's think about it intuitively as you asked:
How many length 3 strings can we make over $\{a,b\}$ where we are allowed to choose $0 b$'s? Well, there's $\binom{3}{0}$ of those: $\{aaa\}$. Hence the first term, $\binom{3}{0}a^3$.
How many length 3 strings can we make over $\{a,b\}$ where we are allowed to chose $1 b$? Well, there's $\binom{3}{1}$ of those: $\{aab,aba,baa\}$. Hence the second term, $\binom{3}{1}a^2b$.
Same as above, but we choose $2 b$'s? $\{bba,bab,abb\}$. To get third term, $\binom{3}{2}ab^2$. This is what Marc van Leeuwen's terse answer is conveying, and is in essence the "combinatorial meaning" you asked for in your example.
Now we choose $3 b$'s? $\{bbb\}$. To get fourth term, $\binom{3}{0}b^3$.
Dave L. Renfro makes a good point:
none of the current answers explain why (a+b)(a+b)(a+b) can be expanded by considering the sum of all possible ordered products of three elements, where the first element comes from the first factor of (a+b), the second element comes from the second factor of (a+b), and the third element comes from the third factor of (a+b).
I'll do my best to explain this one. You see, we're actually repeatedly applying the distributive law of multiplication over addition.
For this example, consider this:
$(a+b)(a+b)(a+b)$
$= ((a+b)a + (a+b)b)(a+b)$ Here you distribute the first factor into the second.
$= (aa + ba + ab + bb)(a+b)$ Distribute again.
$= (aa(a+b) + ba(a+b) + ab(a+b) + bb(a+b))$ Distribute again.
$= aaa + aab + baa + bab + aba + abb + bba + bbb$
Intuitively as we distribute an $(a+b)$ we are "choosing" a or b, but recording both results... in a way.
In effect if we say distribute $c$ over $(a+b)$ we are effectively recording what happens to c when we choose a, and also when we choose b, e.g. $c(a+b) = ca + cb$