To the left you can see a pictographic representation of a typica game play or chain formed in It's Just A Theory (IJAT). The basic process is fairly well explained in this GIF. But we will now detail some of the interruptions you can use to take play.
1) Mentioning Cards
If the current Lead Conspiracy Theorist mentions a concept (either verbatim or strongly suggestive towards) the content of a fact card held by another player, that player may interrupt and take over play. This can be done by either the Lead Conspiracy Theorist saying the card in their explanation, or in response to a question from another player.
Stacey is explaining her theory about why we've never been back to the moon. She mentions spaceships built by the U.S Government and Harry asks who built these spaceships. If she says N.A.S.A (a card that Harry has in his hand). The Harry can play it and takeover. If she doesn't directly say it, Harry might be tempted to put words in her mouth and just interrupt with N.A.S.A regardless and try and take play. If Stacey is fine with this then Harry takes play. If not, then other players should decide whether Harry should take play or not.
in this case Harry may lose his argument and reveal that he has a N.A.S.A card which will mean Stacey will have to be clever with her future responses and story.