In an episode of the first season of ST:TOS, there is trouble between Lazarus from both matter and anti-matter fighting one another. We learn that if matter and anti-matter meet, it would cause the deletion of both universes. Why is this so, as matter and anti-matter meet without destroying the universes (although creating a lot of energy as is the basis of power in Star Trek)?
Your question is very similar to this question, which I answered previously. I think several things from my answer there apply to your question, notably my last paragraph, which states:
"The episode is not very good and does not make very much sense. It doesn't explain itself very well, and much of this is therefore open to interpretation."
It was a poorly written episode - it has been described as "baffling" and "unrewarding" by critics of TOS - and is also poorly acted and edited. In the cast and crew's defence, the original romantic sub-plot was killed due to Southern distributors not wanting a relationship between a white man and a black woman, leading to a last-minute re-write, and the actor originally contracted to portray Lazarus, John Barrymore, didn't turn up on set, forcing the crew to film scenes that did not involve Lazarus until Robert Brown agreed to play the part mid-way through the first week of filming.
Despite this, the theory does make some sense, if you hit your head with a hammer and asphyxiate yourself for a while first. One universe is matter, the other anti-matter. If matter and anti-matter touch, they destroy each other. Thusly, the two universes touching leads to the destruction of both. As to why it would take just one small piece of matter from one universe touching just one small piece of anti-matter from the other to bring about universal annihilation, well, ask Don Ingallis. This is the legend that wrote the original Captain America film - from 1979 - so it's hardly the only bad script he's produced.
The only logical explanation is that they weren't using the terms "matter" and "antimatter" in the conventional sense, or the antimatter that anti-Lazarus was made of clearly isn't the typical antimatter that particle physicists deal with—otherwise he would have been annihilated the moment he stepped foot into our universe unless he had some form of containment field around himself that prevented any air or other particles from touching him. Instead, there only appears to be danger if something in our universe touches its precise counterpart in the anti-universe.
Speaking of which, it doesn't make much sense to call our universe/dimension and the other parallel universe/dimension a "matter" universe and "antimatter" universe in the particle physics sense, as our universe itself contains antimatter, just in very small quantities (though it likely started with nearly equal amounts of matter and antimatter).
So we have to conclude that they're talking about phenomena that are analogous or related to traditional matter/antimatter, but are not quite that. This allows anti-Lazarus to interact with regular people and objects without a matter-antimatter reaction.
And, as the episode explains, the danger comes from the corridor that connects the two universes. It causes Lazarus and anti-Lazarus' ships to be tied to one another. It is also the reason that the universe would be destroyed if Lazarus and anti-Lazarus meet outside of it.
What I inferred from the dialog is that the open corridor created by anti-Lazarus would basically bring the two opposing dimensions together if corresponding objects from each dimension somehow met outside of the corridor. This, in turn, would cause both universes to fully annihilate each other through something analogous to a matter-antimatter reaction.