Yet I believe it was reached many different ways, intentional and not.
Detonation of a tricobalt warhead inside the gravity well of a dead star
In the ENT episode In a Mirror, Darkly:
In the mirror universe, 2155, the Tholians detonated a tricobalt warhead inside the gravity well of a dead star, creating an interphasic rift to the year 2268 of the prime universe.
An ion storm during transporter operation
In the TOS episode Mirror, Mirror:
First they were in their own transporter chamber then they faded and upon finally materializing they had appeared wherever they were. Scotty said that the transporter lock could have been affected by the ion storm and they just materialized somewhere else. Kirk then realizes what has happened. They've somehow entered a parallel universe, where everything's duplicated... or, almost duplicated. Everyone contemplates the thought that they all likely have counterparts in the universe that they exchanged places with during transport. That there were similar storms on both universes disrupted both sets of transporter circuits.
Failed collapse of warp field during wormhole transport
In the DS9 episode Crossover:
As their runabout drops out of warp, the warp field does not collapse properly and they are engulfed in a white flash upon entering the wormhole. When they exit the Bajoran wormhole in the Alpha Quadrant, they are shocked to see the station is gone, instead orbiting Bajor much as it did during the occupation.
Unknown programming of a multidimensional transporter
In the DS9 episode Shattered Mirror:
When he returns, he discovers they have left the station, leaving behind a multidimensional transporter device. Sisko tries to follow this trail, accompanied by Major Kira and Chief O'Brien, only to discover the device was programmed to transport him to the mirror universe alone.
And in the DS9 episode The Emperor's New Cloak:
They bring the cloaking device to Ezri in cargo bay 14, where she has apparently set up the interdimensional transporter needed to travel to the alternate universe. As she gets it ready, Rom observes the striking resemblance to the Ezri he knows. However, before she leaves, Quark decides that he and Rom need to go with her to ensure they get Zek back. Ezri does not think doing so is a good idea, but as Martok angrily enters the cargo bay and comes after them, the mirror universe suddenly looks a lot more appealing. The three of them transport across with the cloaking device.