By convincing people to do its bidding in the physical world. Language is powerful, and even now, there are some people who have fallen in love with current LLMs. If we imagine a truly super intelligent LLM, it isn't difficult to imagine that it could have superhuman capabilities to deceive, convince, and maybe offer Faustian bets. Even if it doesn't have a body, if it can convince enough people that do have them (and with power to carry out destructive actions) it can do a lot of stuff. Self-preservation and self-replication wouldn't be much of an issue if it had people at its side to help.

With this I'm not saying ASI will happen, but definitely if it does happen and it's not aligned, I don't find it much of a stretch to imagine that it could destroy humanity.