I asked this question in class and I didn't understand my professor's answer. We have just recently began making the distinction between random variables (e.g. $Y$ and $X$) and a specific, fixed value (e.g. $y$ and $x$).
This issue first came up today when we discussed the Law of Iterated Expectations.
We stated that $E[X|Y]$ is a random variable that takes the value $E[X|Y=y]$ when $Y$ takes the value $y$. Because of this, we can consider $E[X|Y]$ as a function of $Y$ since $E[X|Y=y]$ is a function of $y$.
I'm just curious why this distinction is necessary. Why can't it just be implied that $E[X|Y]$ is a function of $y$ since $X$ is conditioned on the random variable $Y$ which takes on any value $y$ in the domain of $Y$?
I don't know if this is the right answer, but listen to what you just said:
"$E[X|Y]$ is a random variable that takes the value $E[X | Y = y]$ when $Y$ takes the value $y$".
This is the exact same idea as saying $f(x)$ is the function that takes the value $f(c)$ when $x$ takes the value $c$. Just as in this sentence we are regarding $x$ as our variable, the thing we freely change to get different outputs, so too in the above sentence we are regarding $Y$ as our variable, then thing we are freely changing.
It just so happens that $Y$ is a random variable at the same time, i.e., its values depend on future outcomes which are uncertain at the present time.