Show that an elementary embedding of an L-structure M into an L-structure M′ is an injective function from |M| into |M′|.
I have defined an elementary embedding of M in M' as a function f:|M| -> |M'|, with M,s satisfies an L-formulae if and only if M',f⋅s satisfies the L-formulae.
First, let me restate the definition : $f : M \mapsto M'$ is an elementary embedding iff for every $a_1, \dots, a_n \in M$ and every formula $\varphi(x_1, \dots, x_n)$, we have $M \models \varphi(a_1, \dots, a_n) \Longleftrightarrow M' \models \varphi(f(a_1), \dots, f(a_n))$.
The equality symbol belongs to your language. In particular, for any $a \neq b \in M$, you have $M \models \neg a = b$. Hence, by definition of an elementary embedding, you get $M' \models \neg f(a) = f(b)$, whence $f(a) \neq f(b)$.
We have shown that $a \neq b$ implies $f(a) \neq f(b)$, $f$ is hence injective (as the name elementary embedding suggests).