Well, if I recall it was Trill doctor Bethan Roa's secret serum from the story "Reflections" in The Lives of Dax. The serum's original intention was to boost brain activity in the mentally ill, but it had the unexpected side effect of dissolving symbiosis, and even more unexpectedly of doing so without being fatal to either symbiont or host. This led to a black-market join-for-a-day underground, which was stopped when it was discovered by Jadzia's sister Ziranne.
Bashir remembered the formula and was able to recreate it when the neo-Purist movement used neurogenic radiation to force all joined Trills to reject their symbionts. The radiation was dissolving the symbiosis in a very-much-fatal way, so Bashir made the side effect of the serum into its primary effect and used it to dissolve the symbioses safely, thus saving both the host and the symbiont, both of whom would have died otherwise.
Trill President Lyrisse Maz then took the serum voluntarily in order to dissolve her own symbiosis as an example to her people.
Once the symbiosis was dissolved neurochemically, I imagine it was just the standard surgical operation to physically remove the symbiont from the host body.