They didnt say no checks, they said next to no checks. There actually are relatively few federal restrictions - effectively anyone who is not a criminal can purchase (most) guns legally [0] (some states are more strict, though). Compared to virtually every other country in the world, that is "next to no checks". Many states have no background checks at all for private sales [1], though that wouldn't apply to purchases from a supermarket.

For example, contrast the UK, which in addition to outright banning most firearms such as handguns, requires a "good reason" to issue a firearms permit - and things like self defense are not considered acceptable reasons [2].

[0] https://injury.findlaw.com/product-liability/gun-laws.html

[1] https://consumer.findlaw.com/consumer-transactions/private-g...

[2] https://en.wikipedia.org/wiki/Firearms_regulation_in_the_Uni...