The trick here in "observing a quantum jump" is that this is not equivalent to doing a strong measurement that "collapses" the wavefunction.
The archetypal quantum jump is a two level system with a ground state $\lvert G\rangle$ ("Ground")and an excited state $\lvert D\rangle$ ("Dark") (I'm naming the state kets like they do in the paper to avoid confusion). When we want to "observe the jump", we sit around and wait for a photon to be emitted in order to detect it and say "aha, the system has decayed back to $\lvert G\rangle$!". There's no room for continuous evolution here, when our detector clicks we know the system is in $\lvert G\rangle$.
The paper does something different, a complicated variant of the above that constitutes only a weak measurement saying "the system is currently not purely in the ground state". Whether what they do is meaningfully equivalent to the ordinary colloquial meaning of "quantum jump" I will leave the reader to decide, but its experimental observation is an impressive feat regardless.
Instead of observing the ground state with a strong measurement, they couple the ground state to another excited state $\lvert B\rangle$ ("Bright") via a Rabi drive $\Omega_{BG}$. Then they observe the decay of this excited state to the ground state by a photon counter. The state $\lvert D\rangle$ is explicitly not monitored (hence "dark") and is metastable compared to the bright state, i.e. has a much longer half-time for its radiative decay. So when they observe an emission from the bright state, they know the system just came from the ground state via Rabi oscillation, then decayed again to the ground state - this is the "initial state" of the experiment, we state with the system in the ground state and the ground and the dark state coupled via another Rabi drive $\Omega_{DG}$. Crucially, the Rabi frequencies here are chosen such that the BG transition is much quicker than the DG transition, so it is essentially impossible for the system to get to $\lvert D \rangle$ via Rabi oscillation alone.
So in this setup, we will generally have the detector clicking rapidly, and the system repeatedly being reset to $\lvert G\rangle$ via this variant of the Zeno effect. But because the measurement is not truly continuous, there is a small chance the detector suddenly stops clicking. This is random and not deterministic, so the experiment does nothing to resolve the measurement problem. But since the detector is only coupled to the system probabilistically via the Rabi oscillation to the bright state, this does not constitute a measurement that the system is in $\lvert D\rangle$ - we only become increasingly sure it is not purely in $\lvert G\rangle$ the longer we don't observe a click. This is not a strong measurement, it is a variant of a weak measurement. So when the detector stops clicking, we have some superposition of $\lvert G\rangle$ and $\lvert D\rangle$, but the detector often stops for much longer than one would expect if the system was just Rabi oscillating into $\lvert D\rangle$ - a "quantum jump" has occured, forcing the system into the state $\lvert D \rangle$ much faster due to the interaction with the detector and Rabi drive for the bright state.
When this happens - the detector stops clicking for long enough that the experimenters as satisfied a "jump" is occuring, they do relatively standard tomography on repeated realizations of this state to see in what kind of superposition it is in. They also do a run where they shut off the Rabi drive $\Omega_{DG}$ directly after the jump starts to occur where the tomography results are essentially the same to demonstrate the jump is not driven by the Rabi oscillation. Of course, there is still a random chance that the observation of the bright state might abort the jump and force the system back to the ground state - the entire paper is solely interested in the case where this does not occur.
Since they understand the tomography of the jump well, they also know how to manipulate the state via precise rotation of the state so that the jump "reverses". The time evolution of the system between two detector clicks is entirely deterministic, so this is not a surprise, it's just a neat demonstration. If you look into the actual formulae, you will find that the evolution of the state in this period is directly dependent on the frequency of the Rabi drive $\Omega_{BD}$, so please do not get the impression that this specific behaviour during a "quantum jump" is universal - it is specific to the experimental setup and chosen method of observation.
