I wasn't able to find a definitive proof yet, but most likely it was because of the largest taxicab company (associated with both livery service AND manufacturing of taxicabs) that was called "Checker Taxi".
The 'Checker', particularly the 1956-82 A8/Marathon, remains the most famous taxi cab vehicle in the United States. The vehicle is comparable to the London Taxi in its nationally renowned styling, which went unchanged throughout its use, and also for its iconic status.
In 1940, Parmalee (including Yellow and Checker Cab) became the largest cab company in the United States.
The cars themselves had distinctive checker-marks on the sides, matching the company logo:
I have read unsubstantiated rumors as to the origin of the company's logo to be:
Under Markin, Checker became the first cab company to hire African - American drivers and the first to require that drivers pick up all fares, not just white ones. Some speculate this is why & where the B&W checker stripe originated (src)