Why are the teenage mutant ninja turtles mentored by a rat?
He was a martial art instructor for the Foot Clan, based in Japan, and also had a passion for Renaissance art. One day he was framed by his rival, Oroku Saki (who would eventually turn into his future arch-enemy, the Shredder), for trying to murder their common dojo master. Being unable to prove his innocence, he moved away to New York City, where he lived as a hermit in its sewers and befriended the rats. One day, he came across four baby turtles which were accidentally dropped by a boy through a grate of the sewers. He kept them as pets and treated them like his children. But later, he found the turtles venturing near broken barrels that were oozing glowing pink chemical liquid. As he tried to recover them from danger and attempted to clean them with his bare hands, they were all affected by the chemical which was a mutagen. This mutagen transformed any living being into a mutant whose body became mixed with the DNA of any other living being it was last in contact with. The turtles, being touched by Yoshi, turned into young anthropomorphic humanoid turtles. Hamato himself, having more recently been in contact with sewer rats than his own pet turtles, became a humanoid rat.
Since then, Yoshi raised the turtles by himself, and gave them the names of his favorite Renaissance artists: Leonardo, Donatello, Raphael and Michelangelo, who were to be distinguished by personality and headband color. He also taught them the art of ninjutsu in order to protect themselves better, since people above ground would not accept them.
Hamato was given the nickname "Splinter" by his adoptive sons, due to his proficiency at breaking wooden boards. On one occasion Donatello was able to use the remnants of the ooze that transformed them to restore Yoshi's human self, but the process was only temporary and soon wore off; however, Splinter, having seen how humans could treat each other at times, concluded that he preferred his new state, particularly due to the relationship he had with his sons.