Seeing something before someone else can IS latency. That's why servers on the NYSE floor (or ANY SE floor) are really expensive. Because it gives you an advantage. It isn't humans reacting and buying in HFT, it is computers. The humans are constantly updating algorithms, but HFT means it is the computers doing the reactions (based on algos written).
They have servers in both locations. If you see that a stock starts falling in London you have a time window where you can still sell it at a higher price in say New York, until the price drops to the same level thus closing the opportunity for arbitrage.
How quickly can you "see" the price falling in London and get that message to NYC? Whoever "sees" it first wins, right? That's what they're talking about.
I don't think you understand what many HFT firms do. They arbitrage between exchanges. For example the option or futures price of a security in Chicago and price of the underlying in New York. Also arbitraging between equities listed on multiple exchanges.
Co-location isn't a solution to these problems, and low latency lines between exchanges in different parts of the country and world is a huge huge factor to successfully implementing many/most HFT strategies.
