How did Nikola Tesla invent the induction motor?

Inventions are ideas. He came up with the idea of rotating magnetic fields in 1882, he said. He developed the motor in 1887 and patented it in 1888.

His idea used AC electricity to make rotating fields. Before that DC motors were used with commutators that wore and sparked, to make alternating magnetic fields.

from the Wiki article

In 1887, Tesla developed an induction motor that ran on alternating current, a power system format that was rapidly expanding in Europe and the United States because of its advantages in long-distance, high-voltage transmission. The motor used polyphase current, which generated a rotating magnetic field to turn the motor (a principle that Tesla claimed to have conceived in 1882).

This innovative electric motor, patented in May 1888, was a simple self-starting design that did not need a commutator, thus avoiding sparking and the high maintenance of constantly servicing and replacing mechanical brushes.

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Galileo Ferraris, an Italian engineering professor, conceived the AC motor in 1885. In April 1888, Ferraris published a paper in the Journal of the Royal Academy of Sciences on his motor. In December 1888, Nikola Tesla published a paper on the exact same motor, claiming it as his own invention.

So...how he "invented" the motor was by stealing it from the guy who did.

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