This was -- sort of -- explained in "The Wedding of River Song":
Burning the "body" & (importantly) setting it adrift in a wooden boat concealed the fact that it was the Teselecta & not really the Doctor's body. Not only did the fire prevent anyone examining the "body" closely -- which could have revealed the deception -- but it also gradually destroyed the boat, allowing the Teselecta to sink into the lake, where it was out of sight. Once it was safe from observation, the Teselecta simply walked out of the lake again.
This prevented the Silence from discovering that the shooting of the Doctor had been faked & left them believing that their plan to kill him & to make his death a fixed point in time had worked. The Silence had indeed created a fixed point, which is why River's refusal to shoot caused such drastic damage to the timeline, but the fixed point they created wasn't what they intended. The event that became their fixed point was a fake killing, not a real one.
Because the Teselecta was both fireproof & waterproof, those inside -- including the Doctor -- were unharmed by the process.