Question: Why did the U.S. Navy intend on killing the pirates? Why not just give the pirates their money in exchange for getting Phillips back, and let the pirates go back to Somalia?
Question: How did the pirates manage to recapture Phillips after he tried to escape by swimming?
Answer: They do it exactly as shown in the movie: Phillips, due to currents and his own physical condition, is unable to swim very far from the boat, so the pirates just follow him, trap him, and bring him back onboard. In addition, the Marines fire a flare, which lights up the scene...this was done so they could be certain the escapee was Phillips, but it also makes it easier for the pirates to spot him.
Question: Why didn't the US Navy want the pirates to get Phillips to Somalia? What were they afraid would happen?
Answer: They are afraid that once in Somalia, the pirates would be able to hide Phillips making rescue much more difficult.
BaconIsMyBFFQuestion: How did the pirates trick the crew into thinking they would keep their promise when they make a deal to release Phillips in exchange for giving them their leader back?
Answer: They didn't exactly "trick" the crew, they just never intended to follow through on their promise. The crew has nothing to threaten the pirates with, the pirates have all the guns.
BaconIsMyBFF
Answer: If they pay the pirates, it encourages more to try the same thing. By shooting them it would discourage anyone else from attempting to use a hostage to extort money.