From what I understand, an ordinal $\alpha$ is recursive if it is the order type of a subset of $\mathbb{N}$ that is well-ordered by a recursive relation $\prec$ (meaning, $\mathbb{1}\_\prec:\mathbb{N}\to\{0,1\}$ is recursive). I am mostly happy with this definition. I can see that the ordinals that come out of it are nice and `graspable', that we can construct many of them through standard operations + appealing to fixed points of normal, fast growing functions, though exactly why will be part of my question.

* what is the intuition here? Or in other words, why are we interested in order types of such well-orderings of $\mathbb{N}$? I can see that ordinals like $\omega^{\omega},\,\Gamma\_0$ are nice to have classified in one group, (and are relevant when talking about recursive axiom systems? Not something I know much about though), but I assume there is a deeper motivation that escapes me.
* the construction of these ordinals, through functions like the Veblen function or the $\psi$ function: why exactly are the resulting ordinals recursive? I can see that the ordinals are generated recursively *in the class of ordinals*, but I am having trouble seeing how these functions, defined recursively, relate to the recursive well-orders we want on $\mathbb{N}$/a subset.

For example on the second point, we can define a Veblen function by $\phi\_0(\alpha)=\omega^{\alpha}$ and $\phi\_{\gamma}(\alpha)$ the $\alpha^{\rm th}$ common fixed point of $\phi\_{\beta}$ for all $\beta<\gamma$. $\phi\_{\gamma}$ is thus defined by a recursive scheme. But then why, for instance, does this mean there is a recursive well-ordering of $\mathbb{N}$ of order type say, $\phi\_{\epsilon\_0}(\omega^{\omega})?$ Or if we define $\Gamma\_0$ as the least fixed point of $\alpha\mapsto \phi\_{\alpha}(0)$, how do we know in advance $\Gamma\_0$ is recursive?

The connection is probably obvious but I'm having trouble seeing how this pieces together and would love some help.