Secret Sharing and Multiplication Triples
If you understand that $xy = c \oplus ex \oplus dy \oplus ed$ I am not sure what else is there to understand? 

I wonder if you have understood how the $[x]$ notation works? $[x]$ means that $x$ is XOR shared between between Alice and Bob (I.e., Alice has $x_A$ and Bob has $x_B$, so that $x = x_A \oplus x_B$). When we write $[e] = [x]\oplus[a]$ this that Alice and Bob computes an XOR secret sharing of the value $e = x \oplus a$. Simlarly, if we write [f] = e[x], this means Alice and Bob compute an XOR sharing of $f = ex$. So by extension if $xy = c \oplus ex \oplus dy \oplus ed$ then $[z]$ means we have computed an XOR sharing of $z = xy$.