Asking about compression in this manner it seems you may be over-applying a computer analogy to the brain. The computer model of the brain is an existing resource here that I recommend reading.
As far as this paper itself, it's talking about associative memory in particular, which the hippocampus is highly involved in. The paper you link to is specifically talking about sequence replay, which is where an association (=memory) is built out of a sequence of neuronal activations. For example, you might represent a path through a maze or grocery store as a sequence of spatial navigation steps: moving straight through a corridor, taking a left turn at the milk, following the trail of eggs to the cheese.
In hippocampus, you'll see "replay" of the neural signatures of these sorts of steps. If you look at activity in hippocampus while a rodent runs through a maze, you'll find those same patterns of activity repeat later on when the rodent is just sitting there passively, or while they sleep.
However, these replay events need not take place at the same time scale as the original behavior. You can play this in your own memory, too: maybe it takes you 2-3 minutes to walk into the grocery store and find the cheese, but if you're retracing your steps later you can probably think about the path you took over just a couple of seconds. You also need not think about it in the forward sequence, you can just as well work backwards from your destination and "retrace your steps"; replay in the hippocampus will do this as well.
When these authors talk about compression in ripple events, this is the type of compression they mean: time compression, not a data compression algorithm like Lempel-Ziv. Of course, time is quite a valuable resource in the brain or two a behaving animal, so time compression can also be important for reasons of efficiency, but it's not the sort of thing that necessarily needs to be "decompressed" back to something useful. The associations you might form over several minutes doing some task can be transmitted between brain areas in just seconds or milliseconds. Here's a paper they cite about the "fast-forward" compression:
Euston, D. R., Tatsuno, M., & McNaughton, B. L. (2007). Fast-forward playback of recent memory sequences in prefrontal cortex during sleep. science, 318(5853), 1147-1150.
The paper you link to is a research paper; when learning about a new concept I'd highly recommend looking at relevant review papers, which combine many research results together in a form that's usually more digestible for someone new to the field. Research papers themselves often refer to earlier review papers to ground their results. Here are some of those from this paper that you might look at:
Diekelmann, S., & Born, J. (2010). The memory function of sleep. Nature Reviews Neuroscience, 11(2), 114-126.
Dudai, Y., Karni, A., & Born, J. (2015). The consolidation and transformation of memory. Neuron, 88(1), 20-32.
Eichenbaum, H. (2017). Prefrontal–hippocampal interactions in episodic memory. Nature Reviews Neuroscience, 18(9), 547-558.
and some groundbreaking original papers as well, where you might first focus on their introductions to understand thinking in the field, and then look at their specific results next:
Jahnke, S., Timme, M., & Memmesheimer, R. M. (2015). A unified dynamic model for learning, replay, and sharp-wave/ripples. Journal of Neuroscience, 35(49), 16236-16258.
Karlsson, M. P., & Frank, L. M. (2009). Awake replay of remote experiences in the hippocampus. Nature neuroscience, 12(7), 913-918.
Kenet, T., Bibitchkov, D., Tsodyks, M., Grinvald, A., & Arieli, A. (2003). Spontaneously emerging cortical representations of visual attributes. Nature, 425(6961), 954-956.
Peyrache, A., Khamassi, M., Benchenane, K., Wiener, S. I., & Battaglia, F. P. (2009). Replay of rule-learning related neural patterns in the prefrontal cortex during sleep. Nature neuroscience, 12(7), 919-926.
(paper that prompted this question)
Liu, X., & Kuzum, D. (2019). Hippocampal-Cortical Memory Trace Transfer and Reactivation Through Cell-Specific Stimulus and Spontaneous Background Noise. Frontiers in computational neuroscience, 13, 67.