What does "Chevy Runs Deep" mean?

It's Chevy's slogan.

It means they've been around a long time like, my grandfather drove a Chevy, my dad drove one and now I drive a Chevy too. It's like saying driving a Chevy is a family tradition. That is true to the extent that everybody's grandparents and great grandparents drove vehicles made in the US. There were no imports back then.