I'm not familiar with the DAB system in particular, but I can give you a conceptual answer which I think you will find useful.
In analog radio, things are straightforward. The lower the signal-to-noise ratio, the more the received signal deviates from the transmitted one, and that deviation from ideality in the RF signal results in some closely related deviation in the demodulated audio — the details depending on the modulation used. But, critically, this is on an instantaneous basis; that is, you could plot the original audio, modulated RF signal, signal received with noise, and demodulated audio on a single graph with a common time axis and there would be a consistent relationship. Therefore, since RF noise (that is not deliberate interference) is generally random from instant to instant, so is the effect on the audio.
In digital radio using an audio codec, this time relationship does not hold, so more complex effects are possible because the influence of noise can cause loss or modification of portions of the audio in a consistent way over a longer time interval.
(It would be possible to send a digital audio signal while preserving the time relationship — for example, using a DSD bitstream, or using a modulation with multiple bits per symbol to send an entire PCM sample value at once. However, since it is neither compressed nor error-corrected, it would be very inefficient.)
Generally speaking, an audio codec attempts to represent digital audio using fewer bits than would be taken by the numbers that are the original sample values. This necessarily makes some assumptions about the structure of the signal — that it contains something like voice or music. Therefore, if you put random or corrupted data into the decoding process, it is more likely to generate something that we would call “garble” rather than “noise”, just as a matter of statistics.
Also, digital radio usually makes use of error-detecting (and error-correcting) codes. This means that it is possible for the receiver to know when it is receiving corrupted data, and make a decision about what to do instead. This could be as simple as playing silence instead, but a better option for short interruptions is to proceed using the previous data. Since the decoder is in some sense synthesizing its output, it can just keep doing that, generating the same waveform for longer than intended (as opposed to literally reusing old samples, which creates an obvious glitch as anyone who's listened to a CD skip knows). This is another way you get garble — you're listening to realistic voice sounds being held for longer than makes sense.
Furthermore, according to Wikipedia, DAB uses an “unequal error protection (UEP)” scheme, which means that the error-correction is biased towards preserving aspects of the encoded audio considered more critical.
So your idea of “jumps in bitrate” is somewhat close to the actual situation: it's a change in how many good bits the decoder has available to it, and it compensating as best it can. And in particular, the use of UEP means that a weak signal is more similar to an intentionally lower bitrate than it would be without UEP.