Increasingly, as I deal with Maxwell's equations, I come to the same conclusion. Trying to think them through logically is an exercise in frustration. Stopping worrying about what it "means" and just focusing on the answer is a much easier method to succeed in using them, however I disagree when you say there "is no meaning." It may be extraordinarily complicated and unintuitive, but that doesn't mean it doesn't have any bearings in the physical world. It simply means it's a description.