Representable functions on System T
If every term in system T terminates, then $t\overline{n}\leadsto\overline{m}$ for every numeral $n$, and it is easy to show that a normal form $\overline{m}$ in the empty context is a numeral as well (and cannot be more complex!). In addition, it is easy to show, if the term $t\overline{n}$ is normalizing, that computing the normal form $\overline{m}$ can be done by a Turing machine, which is to say that the function $|t|$ is recursive (the terminology is slightly confusing here). This is what is meant by the first paragraph you have quoted.

Furthermore, every fact stated above can be easily proven in $\mathrm{PA}$, except the normalization of an arbitrary term of system T. They go on to show that each individual term $t$ can be shown to be normalizing in $\mathrm{PA}$ wich they suggest implies that every term $t$ defines a function in $\mathrm{PA}$: for every term $t:\mathrm{Int}\rightarrow\mathrm{Int}$, there is a proposition $\phi_t(x,y)$ such that
$$\mathrm{PA}\vdash \forall x\exists! y\phi_t(x,y)$$

and

$$\mathrm{PA}\vdash\phi_t(\overline{n},\overline{m})$$
iff
$$|t|(n)=m$$

As the following pages explain, the other direction is provable as well, that is to every $\phi$ with the first property, there is some term $t$ in system T that corresponds to it.