The field is represented as a field of polynomials of degree $\leq2$ over the field $\mathbb F_2=\\{0,1\\}$. The notation $(cba)$ means $cX^2 + bX+a$. Since $1+1=0$ the addition table is clear, e.g. $(011)+(010)=(001)$.
For the multiplication, we need to find out what should $X^3$ be rewritten to. From the multiplication table you see that $X^3=X^2 \cdot X^1=(100)\cdot(010)=(011)$ and hence $X^3=X^1+1$ and therefore $X^4=X^2+X$. Now, you can multiply any two polynomials $(cba)\cdot(CBA)$ and rewrite them with degree $\leq2$.
Example: $$(111)\cdot(101)=(X^2+X+1)(X^2+1)=X^4+X^3+2X^2+X+1=X^4+X^3+0+X+1= (X^2+X)+(X+1)+X+1=X^2+X=(110).$$
In general, to find a representation of $\mathbb{GF}(p^n)$ you have to find a polynomial $f(X)$ of degree $n$ that is irreducible over $\mathbb F_p$. Irreducible means that if $f=gh$ for polynomials $g,h$, then either $g(X)=1$ or $h(X)=1$. Then you have that $\mathbb{GF}(p^n)\simeq \mathbb F[X]/f$.