He died after extensive torture and two days after he had been sentenced to death; it was less of an explicit chopping off of the head and more of a withdrawal of medical care after he had been repeatedly subjected to the harshest treatment the Russian state could inflict upon a perceived traitor. Keep in mind that this is the same era that considered disembowelment, being dragged behind galloping horses, and hanging as the warm-up to an execution, and not a method of execution in and of itself, and Russia was noted as being particularly brutal by other Europeans.

 

As to the "Why"; Alexei Petrovitch was alternately disdained and ignored by his father, who felt that he was the weak offspring of a hated marriage arranged by Peter's mother and the conservative faction that she lead at the time. This consevative faction was later taken over by the intensely conservative and Orthodox Tsarina Eudoxia, Alexei's mother. The breaking point came when Alexei attempted to run away from his responsibilities with his Finnish mistress to his brother-in-law, Emperor Charles VI, humiliating Peter on the international stage. Alexei's return marked the start of an inquisition of the conservative faction that he and his mother lead and that had been a perennial thorn in Peter's efforts to reform the backwards nation.

 

TLDR; Alexei was a Jerry. Rick (Peter) managed to off him without crashing the Federation this time.