'Tire' is the American spelling of 'tyre', so 'where the tyre/tire meets the road' is a reference to the crux of the matter, the crunch-point, or the key element of what matters with regard to the problem, in other words.
I believe that it was used in a ?michelin tyres ad campaign as "where the rubber meets the road" The ad. was about relying on the tyre when things go wrong, kind of a politer version of when the .... hits the fan
this is all because in terms of the physics of the thing the point where the tyre meet s the road is the only point where anything has an effect. all the engine and braking power is applied there.