Tokugawa Ieyasu's original domain was Mikawa, roughly where Nagoya is today. After Toyotomi Hideyoshi defeated the Hojo, he offered Ieyasu the eight Kanto provinces formerly controlled by the Hojo in exchange for the five provinces Ieyasu already had, which Ieyasu accepted.

Hideyoshi did this mainly to break the Tokugawa power because Ieyasu had to leave his home province and rely instead on the old Hojo samurai, whose loyalty were questionable. But there were also benefits to this trade. Kanto was relatively isolated, which gave Ieyasu greater autonomy, and very rich, which made Ieyasu the most wealthy daimyo in Japan. With 2,402,000 million koku, he was even richer than Hideyoshi himself. See Elizabeth Berry's Hideyoshi, p. 95-96.

So the answer is Ieyasu didn't move the Shogunate to Edo. He was already in Edo and simply decided to establish the Shogunate there, the same way Yoritomo established his Shogunate in Kamakura because he was already there.

I've posted this here before.

What would become Tokyo did indeed start as a fishing village, located at a strategic crossroads on the Kanto plains. This location would lead to it being fortified, used as a military base, and eventually the base of Tokugawa Ieyasu. When Tokugawa became shogun in 1603, he continued to administer from there, making it the de facto capital of Japan.

Basically, Tokugawa didn't move his court to Edo. Instead, it was not moved from Edo to Kyoto.