There is a lot going on here. I think the quoted text is talking about the ideas of a Popper-like approach to science. But let me sneak up on it.
There is the subject of english meanings of words like "prove." This is quite a bit laxer than a scientific meaning of it. It could be "make plausible" for example, which is not usually what science wants.
There is the somewhat stronger but still pretty lax meaning of experimental efforts. One might design an experiment that explores the predicted consequences of a particular model of a physical system. If the experiment is then performed and agrees with the model, then some might say (with quite a bit of inaccuracy) that they had proved the model. Of course we know this is not correct. Several of our most familiar models stood up to extensive tests with quite impressive accuracy and precision. And yet, they were shown to not agree with later experiments. Newton's gravity followed by Einstein's relativity is but one example.
Experiments can validate a model in a range. The statement of that is kind of dry. "This model works for this range of parameters with this degree of accuracy." It's not proof. It's demonstration the model is sufficient for a purpose to a given level of accuracy.
Very roughly speaking, this is the standpoint of Karl Popper. You don't prove theories through experiment. You test them. They either pass the test or fail. If they pass, you extend the range of validity. If a model has never failed an experimental test it is said to be viable. Meaning it could be the actual description of how the universe works. If it fails a test then it is not viable. Though it could still be adequate for calculation purposes. For many situations using Newtonian physics is perfectly adequate. And it's usually easier to do the calculations that way than using relativity. It's probably not necessary nor helpful to worry about metrics and geodesics when working out the path of a baseball in a game.
Usually scientists will have some version of this process that isn't far from this. There will undoubtably be minor ttweaks and caveats and such.
Note that it places a lot more skepticism on the process than we might in every day life. Compare an idea about, say, your neighbors and their dog and the brown material that keeps showing up on your lawn. If you saw the brown material 8 or 9 times right after doggy's walkies, then you would not keep any doubt about the accuracy of the idea. But we keep trying to test quantum mechanics after it has passed a huge number of really careful and clever tests.
So when the textbook says we don't prove the basic principles it is saying that we don't prove theory. We test it. If it passes the test then we keep the theory as a candidate for being the real way reality is. If it fails we move it over to the "use in this range of parameters for ease of calculation" shelf.
Which is a comfort to many since it tends to produce job security for scientists. A theory is never proved. So we always need to be hunting down places to test it. Can't fire me. General Relativity has not been proved. Quantum Field Theory has not been proved. There is still a huge amount of work to do.