You've already been given most of the entries for the group's Cayley table, so let's just see if we can build the rest of it.
In particular, let $\rho = \sigma\tau = \tau\sigma$ denote the fourth element of the group. Then, just from what you've been given (and the definition of the unit element $e$) we have:
$$\begin{array}{|c|cccc|}
\hline
{\times} & e & \sigma & \tau & \rho \\
\hline
e & e & \sigma & \tau & \rho \\
\sigma & \sigma & e & \rho \\
\tau & \tau & \rho & e \\
\rho & \rho \\
\hline
\end{array}$$
All we need now is to fill in the remaining entries involving $\rho$. Let's start in the corner, with $\rho^2$. Since $\rho = \sigma\tau = \tau\sigma$, we have $\rho^2 = (\sigma\tau)^2 = \sigma\tau\sigma\tau = \sigma\sigma\tau\tau = \sigma^2\tau^2 = e^2 = e$.
What about the rest of the entries? Well, we clearly have $\sigma\rho = \sigma\sigma\tau = \sigma^2\tau = e\tau = \tau$. By similar reasoning, we can also see that $\tau\rho = \tau\tau\sigma = \sigma$, $\rho\sigma = \tau\sigma\sigma = \tau$ and $\rho\tau = \sigma\tau\tau = \sigma$. So the full multiplication table looks like this:
$$\begin{array}{|c|cccc|}
\hline
{\times} & e & \sigma & \tau & \rho \\
\hline
e & e & \sigma & \tau & \rho \\
\sigma & \sigma & e & \rho & \tau \\
\tau & \tau & \rho & e & \sigma \\
\rho & \rho & \tau & \sigma & e \\
\hline
\end{array}$$
Now we can finally check that this multiplication table satisfies the group axioms. Specifically, the closure axiom is trivial to check, $e$ is an identity by construction, and we can see that every element is its own inverse. So all that remains to be verified is that the group operation is associative. (We kind of assumed that it was when we filled in the table above, but that just means that if the operator is indeed associative, those are the values it needs to take for the squares we filled in.)
Since we only have four elements, we could just check all the $4^3 = 64$ combinations by brute force. Or we could be slightly clever and skip all the combinations involving the identity element $e$, since clearly $xy = (ex)y = e(xy) = (xe)y = x(ey) = (xy)e = x(ye)$ for all $x$ and $y$, leaving us only $3^3 = 27$ combinations to check.
Or we could be slightly more clever yet, and observe that the Cayley table we've just constructed is highly symmetric, and in particular that, if $x$, $y$ and $z$ are the three distinct non-identity elements of the group in any order, they will always satisfy the equations $x^2 = e$ and $xy = yx = z$, and thus also all of the following: $$
x(xx) = xe = x = ex = (xx)x, \\
x(xy) = xz = y = ey = (xx)y, \\
x(yx) = xz = y = zx = (xy)x, \\
y(xx) = ye = y = zx = (yx)x, \\
x(yz) = xx = e = zz = (xy)z.$$
This is sufficient to show that the operation $\times$ defined by the Cayley table above is indeed associative for all possible combinations of three elements, and thus $G = (\{e, \sigma, \tau, \rho\}, {\times})$ is indeed a group. (From the symmetry of the Cayley table, we can also clearly see that $\times$ is also commutative, making the group abelian.)
Of course, arguably the easiest solution is simply to recognize that the Cayley table we've constructed above matches that of the Klein four-group, and that the putative group we've just constructed above is thus isomorphic to it. Of course, you have to be aware of the existence of the four-group (and, in particular, of the fact that, up to isomorphism, there are only two groups with four elements — the other one being the cyclic group of order 4, which your group obviously isn't) to make use of this particular shortcut.
Best Answer
Let me try to guess what can have come into their mind.
First of all, as mentioned in the comments, these were found in order to mimick the properties of $(a,b) \mapsto ab^{-1}$ in a group.
To get a group from the single operation $/$, we need to recover the neutral element. Bit $e= aa^{-1}=a/a$ for any $a$. So to get $e$ we need to have $a/a=b/b$.
If we call $e$ the common value of $b/b$, we need it to be a neutral element : $a/e = ae^{-1}= ae= a$. Hence $a/(b/b) = a$.
We need $e$ to also be neutral on the left : we need to say how $e/(b/c)$ relates to $b/c$, but since there is an inversion in the process, we need to put $c/b$.
At this point you have recovered the neutral element, and part of the inversion. Still you have no law concerning general $a,b,c$ that can account for associativity.
This is where 4 comes in. It relates a general $a,b,c$ and in terms of $ab^{-1}$ is : $(ac^{-1})(bc^{-1})^{-1} = ab^{-1}$. This means (thanks to 3. and to what it tells us about $(fg)^{-1}$ ) $(ac^{-1})(cb^{-1}) = ab^{-1}$. So this is a version of associativity in terms of $/$.
Now we have recovered the neutral element, the inversion, and associativity : we've got a group.
Note that analyzing where these laws come from tells you a great deal about how to prove that $(G, *)$ is a group, because you now have a clear route.