Why must energy transitions in hydrogen atom be between stationary states?
Why must this transition be to another stationary state, which has definite energy?
It does not! The OPs question is really an objection to the textbook presentation.
Atomic transitions are a dynamical process
When a photon or, more generally, a wave packet/driving pulse hit an atom, they set of dynamics of the electrons. In general, these dynamics are time dependent.
When you have very short pulses, you can study or even control the dynamics that happen. To show you that people do that in current research, here is a recent representative reference, where a transition of an atom is observed and controlled in the time domain: Science 354, 738 (2016)
The eigenstate/stationary state picture is useful
Nevertheless, the energy eigenstates of the atom are often still useful, because they can be used as a basis to represent and understand the dynamics. When a weak pulse hits an atom, we can often describe it in terms of a time dependent quantum state $|\psi(t)\rangle$, which in turn can be decomposed into its eigenstate contributions $$|\psi(t)\rangle = \sum_{n \in \mathrm{eigenstates}} c_n(t)|\phi_n\rangle$$ with time dependent coefficients $c_n(t)$.
This is also what the textbook treatment aims at. In addition, in perturbation theory (for weak pulses) one may actually have transitions between stationary states: The atom is in an eigenstate, a pulse kicks it, the atom is in a different eigenstate. The latter effect can, for example, be observed as absorption lines, see annav's answer 1. To understand these phenomena, the textbook treatment is immensely useful.
The transition picture can break down
This transition picture can also break down. For example, when you have very strong laser pulses, the many photons hitting the atom destroy the eigenstate structure, such that the above mathematical description becomes less useful. Essentially, you can have the whole atom flying apart and if you wanted to write the wavefunction for that in terms of eigenstates, you would need a lot of them, which makes the picture less useful.
1 Edit: In fact one can see that the transitions are not fully discrete in the absorption spectra - the absorption lines have a width, which indeed corresponds to the inverse of the life time of the transition. This observation hints at the dynamics taking place, in this case do to decay processes.