From the Memory Alpha article on the episode:
Commander Sisko tells [O'Brien] that one of the plasma conduits blew out while he was attempting to re-route a phase inducer. Bashir tells him that he has a mild case of radiation poisoning, before injecting him with a dose of hyronalin to counter the effects...
Dax theorizes that the ionizing radiation he was exposed to earlier may have something to do with [his time travel]...
Bashir theorizes that the singularity is attracted to the delta series radioisotopes in O'Brien's body, effectively pulling him along like a magnet. And if that's the case, he may have a cure for it, but it's going to take time...
So, the quantum singularity in the Romulan warp core was pulling O'Brien through time via an interaction with the radiation he was accidentally exposed to. The staff was able to control the travel by moderating O'Brien's radiation levels. An extra dose of radiation allowed him to make his last jump, but:
[Bashir] explains to [O'Brien] that the [radiation dosing] device is already calibrated for the return trip so all he has to do it press it again to return to the present. Bashir warns him that he cannot wait too long or his body will fail due to severe radiation poisoning.
Having made his jump, and with events proceeding to a fatal conclusion,
The future O'Brien realizes the orbiting quantum singularity was the power source of the Romulan ship and tells O'Brien to go back and stop it. But he can't. The radiation poisoning has taken its toll on his body and he would die if he did. Instead he takes off the device and hands it to future O'Brien, convincing him to go back in his place.
As for creating a time machine of this phenomenon, there are some in-universe barriers:
- The radiation dose needed for a jump of less than a day was fatal without immediate treatment. This reduces the applications considerably.
- One would have to have access to a quantum singularity. Fine for the Romulans, no one else has been shown using that technology.
- One would have to be able to simulate O'Brien's accident closely enough to replicate the effect. This could involve a lot of trial and error.
- There are many other paths to time travel in the Star Trek universe. The slingshot effect may or may not be possible, if not practical, for anyone with warp drive. Someone determined to time travel would probably seek a method which doesn't involve radiation poisoning.