I am sure nicer answers will be posted, but I will post my attempt at explanation, though I am a group theory amateur myself. (Sorry for the extremely long answer!) Also check the wikipedia page for more details and for precise definitions/theorems.
This is an example of a "presentation" of a group. A few definitions will be useful for the discussion. The elements $a$ and $b$ are called the "generators" of the group, and the "equations" $a^2 = e$, $b^3 = e$ and $ba = ab^2$ are called "relations". A "word" is a string involving the generators and their inverses (e.g.: $a^{-2} b^{-4} bab^2$). Note that it is sometimes straightforward to rewrite a word in terms of just the generators. For instance, in this example, we can rewrite $a^{-1}$ as $a$ and $b^{-1}$ as $b^2$; when we do this $bab^2 a^{-2} b^{-6}$ becomes $b a b^2a^{2} b^{12}$.
Completely determining the multiplication table
Yes, it means that the product of two elements can be expressed as one of these elements themselves. This can be done by simplifying the product by repeatedly using the relations $a^2 = e, b^3 = e, ba = ab^2$. (In simple examples such as this, you can usually do routing group operations just by inspection.)
Where are the inverses?
Inverses are automatically and implicitly defined by the relations that you have prescribed. For instance, $a^{-1} = a$ and $b^{-1} = b^2$. Why? Because the relation $b^3 = e$ implies that multiplying $b$ by $b^2$ (to the right or to the left) gives you $e$. So, $b^2$ must be the inverse of $b$. You can similarly check that $a^{-1}$ is $a$. (@WillO's comment that you can always write $b^{-1}$ as a power of $b$ in a finite group is relevant here.)
To compute the inverse of words (like $bab^2 a^{-2} b^{-6}$), you can use the identity $(xy)^{-1} = y^{-1} x^{-1}$. For example, we have $(bab^2 a^{-2} b^{-6})^{-1} = b^{6} a^2 b^{-2} a^{-1} b^{-1}$. This can, of course, be simplified further using the usual $a^{-1}=a$ and $b^{-1} = b^2$ trick.
Why 3 equations? In one sense, $3$ equations is not sacred. You can take any set of generators and any set of relations between the generators, and you will end up with some group. For instance, taking the generators to be $a$ and $b$, and the set of relations to be the empty set, you will get what is called the free group on $\{a, b\}$.
But not all examples that we obtain this way are nice; for instance, the group may turn out to be infinite or even uncountable. If you want to express your finite group in terms of a presentation, that is also quite easy. We will come to this point later.
Why is a multiplication table significant?
Pedagogical value. A multiplication table is a proof that the group that the author defines is in fact a group. I remember that when I first read about representing groups using relations, I used to hand-compute the whole table, and check the group axioms. Though tedious even for medium-sized groups, the exercise helped me intuitively understand the idea of group presentations and also taught me some simple tricks in simplifying the products of elements of a finitely generated group.
As a finite presentation. The multiplication table of a finite group $G$ also automatically gives you a finite presentation of the group. (I found this point in the Wikipedia page while researching for writing this answer.) Simply define the elements of $G$ as the generators, and for each entry $g_i g_j = g_k$ in the multiplication table, add a relation $g_i g_j g_k^{-1} = e$. (In this sense, finite presentation is a generalization of finite number of elements.) But of course, out of the $|G| \times |G|$ relations that we can write this way, only a few are useful; the remaining can be deduced from other relations. For example, you show how to deduce the relation $(ab^2)(ab^2)e^{-1} = e$ from the usual relations $a^2 = e$, $b^3=e$ and $ba=ab^2$.