The movie explains this in a very hand-wavy manner. The omega mimic can control time, this power can be controlled by the high level mimics. If bathed in the blood of the high level mimic, one essentially gains control of that power, and can loop things backward in time. Rita indicates that when she had this power she wasn't able to control it, it simply triggered on her death automatically.
This, of course, causes continuity problems, because in the second, and many other later loops, the main character wasn't bathed in the blood of the higher level mimic, so if this really is a form of going back in time, then, physically, the main character never obtained the power a subsequent loop, and the mimics should have won merely by restarting time prior to that power transfer.
The manga hinted at what was going on, but the book delved into it further.
Rather than a form of time travel, or reversal of time, only memories were transferred from the future. This implies parallel universes.
The main character lives the first day, but by killing the strange mimic they gain the ability to "radio" (tachyon waves) their memory back to their prior self . There is a distinct "beginning" time, though, that can't be altered - the memories are sent back to the person at one point in time, so one can't send memories further back.
So essentially a hundred or more parallel universes split off from that point in time, each one sending back the memories created from that universe to the original character. The character's body, therefore, and brain receive all these memories, and it appears to them that they've lived them sequentially, but the reality is that they haven't at that point. Their brain is just trying to make sense of it. In the book/manga this results in headaches, the more memories are given. In the movie there's "visions" and nose bleeds.
Note, also, that the manga indicates the mimics are actually terraforming bio-robots with little intelligence of their own. If you think of memories as merely computer data being radioed back via tachyon pulses (which are theorized to travel back in time) and robots of the past receiving and acting on future data, it might make more sense. The difficulty is accepting biological robots with this capability, and further accepting that humans are biologically compatible enough to also be turned into receivers and transmitters of the same method.
I don't know that there's a good explanation for how things work in the movie, but you could adapt the same kind of technique. The omega mimic can send memories back, once connected, the connection only grows stronger and it can't resist sending memories of its own back. The characters could be wrong about whether it's time travel, or simply memory sending.
In the book it's the fact that the human killed the important mimic - the mimic somehow imprints on the human. Perhaps the extraordinary strong tachyon pulse it emits on death is involved in getting the human to send one of their own back to their own biological body in the past. Note also that the human doesn't die at this point, he continued to fight, and eventually died from another mimic.
At any rate, to the person experiencing it, it feels like time travel. In the book and manga, it's clearly not time travel, but simply sending memories back.
Since both stories are told from the point of view of the main character, and their understanding is obviously limited, then it is likely that we can theorize forever and never come to a real understanding in either case.