Why use maximum likelihood for word prediction?
That approach is using a score function (in a particular way).  It is also using maximum likelihood.  Both ways of describing the approach are valid.

Why is it useful to think about this in terms of maximum likelihood?  Because it gives a more systematic way to think about this problem and others like it.

If we just want to think about maximizing a score function, the question that raises is -- what score function should we use?  How do we choose one?  It sounds arbitrary.

In contrast, the maximum-likelihood approach gives us a principled way to choose the function.  Here is how to derive that formula, given the maximum-likelihood principle.  First, they have apparently decided to use the following approximation for the probability of seeing $w_t$ as the next word, given history $h$:

$$P(w_t,h) = \exp(\text{score}(w_t,h)).$$

Once you've decided to do that, then by the definition of conditional probability it follows that

$$\begin{align*}
P(w_t|h)
 &amp;= \frac{P(w_t,h)}{P(h)}\\
 &amp;= \frac{P(w_t,h)}{\sum_{w'} P(w',h)}\\
 &amp;= \frac{\exp(\text{score}(w_t,h))}{\sum_{w'}\exp(\text{score}(w',h))}.
\end{align*}$$