By taking over decision-making roles, and then making locally-optimal decisions without regard to negative externalities. (Humans are pretty good at this already, but computers can streamline that whole cumbersome process of making bad decisions.)
The world is already to a significant degree run by machine learning algorithms that are designed to maximize shareholder value in some way, and many of them are deliberately engineered to manipulate the public, often by using their personal information.
Now, consider if this assortment of for-profit AIs were able to replace humans at the top of the decision-making chains in their respective organizations, and then were able to bribe/blackmail/manipulate the political and social structures of society to increase their wealth, power, and influence. It might seem kind of silly to consider computers doing this, but it's at least sort of how the world works now with humans in charge. If AI was in charge, it removes the restrictions of empathy, moral principle, and mortality on the acquisition of wealth, not to mention limited time and attention.
(Why would we put AI in charge of corporations, you might be wondering? How many boards of directors would disregard the idea if such an option could be reasonably expected to increase profits? And how many middle-class workers would refuse to invest in such companies if they had the best dividends and stock value growth?)
So, maybe we end up in a profit-centered dystopia where computers own all the wealth and people are effectively slaves. That's not the end of mankind, but it puts us in a position of no longer controlling our own destiny and being unable to react to existential threats. For instance, we might not be able to do anything about climate change because our AI overlords don't individually see any advantage in spending resources on that.