What does "maximum value" of a set of random variables mean? In our statistical mechanics lecture the professor said something along the lines of:

If we have some independent random variables $x_1,x_2,x_3,...,x_n$ having identical distributions:
Suppose $M_{n}=\text{max}(x_1,x_2,x_3,...)$, then we say that
  probability that $M_{n}<x$ is $\text{$\text{Pr}(M_{n}<x$)}$ (say).
In such a case
  $\text{$\text{Pr}(M_n<x$)}=\text{$\text{Pr}(x_1<x,x_2<x,x_3<x,...$)}=(\text{Pr}(x))^{n}$

Now, first of all I don't understand what he meant by $M_{n}=\text{max}(x_1,x_2,x_3,...)$. What does maximum of a set of random variables even mean? Does it refer to the random variable which can take the highest value?
Secondly I don't understand the step $\text{$\text{Pr}(M_n<x$)}=\text{$\text{Pr}(x_1<x,x_2<x,x_3<x,...$)}=(\text{Pr}(x))^{n}$

Yes, it is the random variable with the highest value.
For your second question, consider $n$ real numbers $x_1, \dots, x_n$. Suppose the largest of these is less than some value $x$. Then it follows that $x_1, \dots, x_n$ must all be less than $x$ as well. By independence
$$\mathbb{P}(M_n < x) = \mathbb{P}(X_1 < x, X_2 < x, \cdots , X_n < x) = \mathbb{P}(X_1 < x) \mathbb{P}(X_2 < x) \cdots \mathbb{P}(X_n < x)$$
and assuming that $X_1, \dots, X_n$ are all drawn from the same distribution (this was not stated in your question), it follows that
$$\mathbb{P}(X_1 < x) \mathbb{P}(X_2 < x) \cdots \mathbb{P}(X_n < x) = [\mathbb{P}(X_1 < x)]^n$$
because all of $\mathbb{P}(X_1 < x)$, $\mathbb{P}(X_2 < x)$, ..., $\mathbb{P}(X_n < x)$ would be identical.
