Typically when studying some class of mathematical objects, we study them together with some kind of corresponding notions of substructure and with "reasonably-behaved" maps between them. (Such as groups, subgroups, homomorphisms; vector spaces, subspaces, linear maps; topological spaces, subspaces, continuous maps.) This is especially natural from the viewpoint of category theory.
Usually when dealing with well-ordered sets, initial segments are more suitable than arbitrary subsets. (For example, comparison of ordinals is usually defined in terms of isomorphisms with initial segment. When we have a chain of well-ordered sets and we want to get again a well-ordered sets - which might happen in some arguments based on Zorn Lemma - this does not work for ordering by inclusion, but it does work if we use initial segments.)
After gaining some experience with well-orders, it is possible to accept the fact that this is a reasonable thing to do, simply because it works. However, is there some insight into definition of well-ordering which would lead more directly to the fact that initial segments are "the right version" of substructure in this case?
This seems to me to be related to the question what is a reasonable "morphism" between two well-ordered sets. I am not sure what would I choose as morphism if I wanted to work with category of well-ordered sets - maybe a reasonable choice could be something which behaves reasonably well w.r.t. successsors and suprema? (I.e., something similar to normal functions; only if we want to include also the case that the map is not injective, we would probably have to modify this notion slightly.)
TL:DR; Is there some perspective from which initial segments (rather than arbitrary subsets) of well-ordered sets can be immediately seen as the most natural version of substructure/subobject for well-ordered sets?
Theorem For every well-order $(X; R)$, where $X$ is the universe of the well-order and $R \subseteq X \times X$ is the actual well-ordered relation, there is a unique ordinal $\alpha$ with a unique isomorphism $$ \pi \colon (X; R) \to (\alpha; \in). $$ In fact $\pi(x) = \{ \pi(y) \mid y R x \}$ for all $x \in X$.
We call $\alpha$ the order-type of $(X; R)$ and in reality we don't care about specific well-orders - we only care about their order types. (Much like in group theory we only care about groups up to isomorphism.)
Lemma Let $(X;R)$ be a well-order and let $Y \subseteq X$ be any subset. Then $(Y; R \restriction Y)$ is a well-order.
Corollary Let $\alpha$ be an ordinal and let $Y \subseteq \alpha$. Then $(Y; \in)$ is a well-order and hence there is a unique ordinal $\beta$ with a unique isomorphism $$ \pi \colon (Y; \in) \to (\beta; \in). $$ It's easy to see that $\beta \le \alpha$ (and $\beta = \alpha$ iff $Y = \alpha$), so that in fact $(Y; \in)$ is canonically (uniquely) isomorphic to an initial segment of $(\alpha; \in)$.
The same argument applies to any subset of any well-order - by pushing the subset pointwise into its order-type.
This is why we only care about initial segments of well-orders - in fact we really only care about order types of well-orders and hence only about ordinals.