Understanding Evaluation Mapping

Solution 1:

We have (presumably) some set of functions $f_\alpha: X \to \Bbb R$ where $\alpha$ ranges over some index set $A$.

The set $P:=\prod_{\alpha \in A} \Bbb R$ indeed is the set of all functions from $A$ to $\Bbb R$. So an element of $P$ is just a function. But I think part of the confusion lies in the fact that if $g \in P$ authors use the notation $g_\alpha$ and $g(\alpha)$ interchangeably. They mean exactly the same though: the value of $g$ at $\alpha$. The first notation is more "product"-like (like sequence indices if we the functions we defined on $\Bbb N$), the second more "pure", just function-notation.

The evaluation function $h:X \to P$ is just the function that maps $h(x)$ for some $x \in X$ (now constant for a moment) to a function $h(x) \in P$; in order to define that function we have to define its values at every $\alpha \in A$ and this is done in an obvious way:

$$h(x)(\alpha)=f_\alpha(x) \text{ or in the other notation: } h(x)_\alpha = f_\alpha(x)$$

So either way you're just defining the function $h(x)$ for temporily fixed $x \in X$. With $\alpha$ we only have the function $f_\alpha$ from our family at our disposal and with a point $x$ and a function, what else could you do but evaluate that function in that point?

Hope this helps. It's indeed just a function notation in a different form. It's good to be aware of it as many texts use this.