That Italian restaurants became linked to those tablecloths might have began in New York before Prohibition, where there were many more Italian restaurants than elsewhere. The Italian restaurateurs could have looked to the city’s popular, casual French bistros, or German, or affordable restaurants of any or nearly every stripe – except for Chinese, it seems – in the city for an example to borrow to suitably cover their tables. The starched white tablecloths at the nicer establishments signified something classier and more expensive. And cleaning and ironing those white tablecloths were more expensive, too. The affordability of these red-and-white-checkered options were certainly a significant factor, too. And, conveniently and maybe importantly, that busy red-and-white pattern help obscure stains from red sauce and red wine, which were brought to about every table.
As Italian-themed restaurants became more ambitious and pricier, and maybe more Italian, these lost the red-checkered patterns. These were seen by many diners as old fashioned. And that is reason is why you will still see them at some intentionally homier Italian-American places. Italian-American not Italian.