They can front run because they receive all of the order flow decimals of a second before the orders are executed. Let's say for example you saw (or your computer told you) that X market orders for Y shares to buy a stock just entered the market...your program tells you this is a lot of volume for this stock and it should push up the stock. Before any of the orders are executed your algorithm jumps in front and buys the stock and sells it a second later for a nice profit.
This is a very simplistic example.
That's the whole point...Michael Lewis said he can't believe it hasn't been made illegal....5 years ago, during the financial crisis there was a good story on it in the New York Time and one of the Senators said they were outraged and he was going to change it, etc, and of course nothing ever happened.
The bigger story of HFT is that it enables the big banks to manipulate the stock market. Because they know all market orders before they are executed, they are able to cancel all bids and offers before it is possible for anyone to hit or take them. They can also put up huge orders on the bid or offer to make a stock look weak or strong in the short run, knowing there is no risk of being executed. If they see another order come in they can just cancel their order.
The most criminal advantage HFT gives them is that they can front run orders. Let's say for example that Fidelity calls Goldman Sachs and says they want to buy 1 million shares of Tesla. This information is priceless as Tesla stock is guaranteed to fly higher on this trade. Acting on this information is the most blatant form of insider trading that exists. However, somehow the big banks can create programs that do exactly this. Whenever they see a huge influx of buy or sell volume they can quickly jump in front and close out the transaction a few seconds later for a huge profit.