I can reassure you that this property is not immediately self-evident. In trying to describe/enumerate the set of normal forms, the main observation required is the following:
Abstraction preserves normal forms: if $t$ is normal, than so is $\lambda x.t$.
Application does not preserve normal forms: if $t$ and $u$ are normal, $t\ u$ may contain a redex!
We want to characterize the normal forms for which we can not create redexes when performing applications. Obviously, this occurs iff $t$ is a $\lambda$-abstraction. In particular, we can take $u$ to be anything we want, so long as it is in normal form.
For $t$, we need either a variable or an application, itself already in normal form. This allows a convenient recursive definition for $t$, which we will call neutral:
with $t_2$ any normal form, and $t_1\ t_2$ also in normal form, i.e. $t_1$ also a neutral term.
But this is exactly your definition of
One can carry out this process further to characterize exactly the normalizing (resp. strongly normalizing) terms, if we furthermore allow weak head expansions.