Trying to understand interpretation and denotation in FOL
Consider the expression $+(x,\times(y,y))$, which is the FOL way to write $x+(y\cdot y)$. If I substitute $x=3$ and $y=5$ and do the arithmetic over the integers, what do I get? I need to evaluate the expression recursively.

In order to evaluate $+(x,\times(y,y))$, I need to evaluate $x$, to evaluate $\times(y,y)$, and then to combine them using the function $+$.

Evaluating $x$ is easy, since $x$ is just a variable. My assignment says that $x=3$, so the value of the expression $x$ is $3$.

Evaluating $\times(y,y)$ is more difficult. First I need to evaluate $y$ (twice) - as before, this is easy, and gives the result $5$. Now I need to combine the two values $5,5$ using the function $\times$. I have a large table which, for every pair of integers $a,b$, tells me what $\times(a,b)$ is. Looking at the table, I find out that $\times(5,5) = 25$.

Now that I have evaluated $x$ To $3$ and $\times(y,y)$ to $25$, I can finally look at the table for $+$ and deduce that $+(3,25) = 28$. So $28$ is the value of the entire expression.

The formal definitions you quote in your question attempt to capture the preceeding process in an abstract form. It's best if you work out the correspondence on your own.