“Take the next left, cut over to the highway.” What does “cut over” exactly mean?

go onto/enter the highway. i’m guessing you take a left, then there’s a highway entrance you take to “cut over” to the highway. “cut over” means move over to another spot, often in sort of an abrupt manner. in this case, entering/moving onto the highway

you're changing lanes (cutting in front of other cars). this expression is particularly common if you don't have a lot of time/space in which to do so.

Cross over the (roughly) parallel road. You're on a road that runs alongside the highway. You'll turn left, and go straight across the highway, without driving along it.