Salt increases intravascular osmolarity, which increases water retention to correct the osmolarity. This is freshman inorganic chemistry. There's plenty of biomedical science behind renal pathophysiology and the link between salt and blood pressure is as close to an if-then statement as there is in medicine. We use all sorts of drugs and methods to play with this central mechanism and can observe the effects in real time at the bedside.

It's not "science". It's science, as in, emperically demonstrated, over and over, billions of times, every time, science.

That said, healthy people manage their osmolarity and blood pressure through other renal mechanisms. The people who run up against this salt-blood pressure issue are generally medically fragile, older people who have outrun all their other compensation mechanisms. The question here is more about the cost-benefit of pushing this particular public health message. The vast majority of healthy people don't benefit, and the sick people get the word from their physicians anyway.