Generally these tables are interpreted as taking an $x$ from the left column and a $y$ from the top row and putting its product $xy$ in the $(x,y)$ position.
You have already been told that $d$ goes in the $(a,b)$ position, and that $e$ goes in the $(c,a)$ position, and that $b$ goes in the $(d,c)$ position. Since groups of prime order are abelian, you can also conclude what $ba,cd$ and $ac$ are.
Since $a,b,c,d,e$ are likely assumed to be distinct, you can also tell from these that the only candidate for the identity is $e$, and so that allows you to fill in the last column and the last row rapidly. At this point you also learn that $a$ and $c$ are inverses of each other.
Using that relationship, you can deduce from $ab=d$ that $b=cab=cd$, so another entry appears in the $(c,d)$ position. As you get further along, you should be able to deduce each position.
Don't forget also that you have another tool at your disposal, namely that all the elements satisfy $x^5=e$. Another thing is that $a,c$ are paired up as inverses, and $e$ is its own inverse... what can you conclude about $b$ and $d$? Also, show that $a^2\in\{b,d\}$: if you try both of them out, you should see immediately that only one is consistent with the relations.
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In number $4$ we have $da=d$ and $de=d$. Now in a group, $d^{-1}$ exists, and it follows $a=e$ (from $da=d=de$), a contradiction. Hence it is not a group.
Best Answer
While applying group theory (and other aspects of abstract algebra) can be concrete, in general we don't care much about what $e, g$ and $\cdot$ actually are. They are two elements and a multiplication, and how the multiplication works is defined by the table. Then when you come across a concrete example that works like this, you can think "Oh, I know this group" without ever having seen those exact objects before. This is the power of abstraction.
So you can ask about whether this group appears in any kind of concrete, not-too-farfetched setting, and that's a perfectly fine question to ask. Actually, having a number of different representations of each of the most common groups is a good way to spot connections between mathematical objects that isn't at first obvious.
In this case, you have $\{1, -1\}$ with regular multiplication. You have the set of linear isometries of a line with composition as the operation. You have the subset $\{0^\circ, 180^\circ\}$ of rotations of the plane (or of a circle), with composition as the operation. You have the set $$ \left\{\begin{bmatrix}1&0\\0&1\end{bmatrix}, \begin{bmatrix}0&1\\1&0\end{bmatrix}\right\} $$ of matrices with matrix multiplication. You have the additive group of integers modulo $2$. And loads of others.
That being said, don't be afraid of the purely abstract notion either, because at times it is going to be the only thing you have. And if you're not used to working with it at that point, it's going to be tough.