Because some jackass had to go and ruin it for everyone else.

Cough

According to Edo bakufu records, in the night of Shōhō 3.III.25-26 (May 10-11, 1646) somebody flew a fire-lit kite and it fell within Edo castle. Like all pre-modern cities, Edo was prone to fire and most of the city would in fact burn down (including the castle's keep) eleven years later. So the bakufu was understandably quite pissed and ordered kites banned.

Funny enough the Ryōei Hinamiki (above) seem to think a kid flew it, but when it was cited in the Gojiki the entry's changed to "someone" so they might have never found out who flew the kite and whether it was a deliberate arson attempt or not.

Also the 1646 order was obviously not followed, because the order was issued again in 1649, 1656, 1659, and 1660, all of which say "as already ordered previously." The wording of the 1646 order doesn't seem to have survived, but interestingly the 1649 order seem to ban all kite flying, while the subsequent orders specified it was children who were not allowed to fly kites. But the 1656 and following orders banned the sale of kites as well. Also of note is that all orders specified the ban was only "within the city". And the orders haven't been found in Kyōto or Ōsaka. The fact that the bans were only for children in Edo suggest that the bans, at least the third one onwards, was also to prevent children damaging stuff, getting hurt, getting in the way of adults, and just in general causing raucous for the many high-ranking samurai in Edo.

Afterwards the order was either followed enough to please the bakufu, or the bakufu gave up. The next surviving ban order was issued in 1748, but drawings and mentions of kite kept appearing in the meantime (and hence forth like this one stated to be in Edo, this, and this).

Finally, here's a popular story. The modern pronunciation for kite in Japanese is tako, which means octopus. However, in early- and mid- Edo poetry they're called ika. Ika means squid. Both probably liken the ribbons and strings attached to the kites to the animal's legs. The 1649 and 1656 ban orders though call them tako. The story is that the people of Edo tried to get around the ban by saying "we're not flying squids, we're flying octopuses", forcing the bakufu to issue the orders to ban tako. The more likely explanation though, and supported by Edo sources, is simply that, at a time before standardized language and education, these are simply regional variations of the name. Ika was the name for kite in the kinai, and tako was the name in the kantō, and other regions had their own names.