You get "definite answers on the laws" from a tax lawyer, not from anonymous sources on the Internet. None of us here are lawyers. So what follows is not a "definite answer" and it is not legal advice, but it will get you started on what the considerations are.
That said, yes, the general rule is that you have to pay tax to any state that you work in, on the income that you earn for working in that state. You would have to file a nonresident tax return in each state that you worked in. Some states might have thresholds such as a minimum amount of income in the state or a minimum number of days. You would have to research the rules for each state that you work in. The best approach would be to consult a tax professional who specializes in multi-state taxation.
But your having to file and pay tax in multiple states is not your employer's problem. The problem for your employer is that some states will consider your employer to have a presence in that state if they have even one employee (you) working in that state. That means that the employer would have to withhold state tax from your pay, the company might have to file a tax return in the state and pay state tax on some of its income, and it might be subject to other rules for employers in that state. Essentially it would be the same as if they had an office in each state that you worked in. Again, some states might not require this if the amount of time you spend in the state, or the amount that you earn, is below a certain amount. But your employer understandably would not want to deal with all of this if they do not already have a presence in other states that you work in. An additional burden for the employer, of course, is that they would have to keep records of how much you worked in each state
