Anti-Lazarus' civilization discovered a way to slip through the alternative warp that connected the two universes. When he made this trip, his existence was revealed to Lazarus, who became insane and obsessed with his destruction. Lazarus pursued his counterpart across time and space, eventually arriving in 2267, where their conflict drew the attention of Starfleet and others – because it threatened the existence of both universes.
After learning the full story, Captain Kirk assisted Anti-Lazarus by forcing Lazarus into the corridor and destroying Lazarus' spaceship. This action sealed the corridor and saved the universes, but left the two men trapped together for the rest of time. This was a price Anti-Lazarus was willing to pay to save two universes. (TOS: "The Alternative Factor")