The comments is completely wrong, and that is why questions should not be answered in comments, wrong answer in comments can not be downvoted.
The covariant form of Maxwell's equations:
$$\partial_{\alpha}F^{\alpha\beta} = \mu_{0} J^{\beta}$$
$$\partial_{\alpha}F_{\beta\gamma} + \partial_{\beta}F_{\gamma \alpha} + \partial_{\gamma} F_{\alpha \beta}=0$$
are indeed Lorentz invariant, in particular you wrote them in a way where you know how everything transforms.
But Maxwell tells you how the fields evolve, given the charge and current. It doesn't tell you the motion of charges. For that you'd need something completely different, like the Lorentz Force Law and some relativistic version of Newton's Laws.
Griffiths is saying that if you write down Maxwell in any frame then you can use that (to find the field and then e.g. use the Lorentz Force Law and $\vec F=d\vec p/dt$) to find the kinematics of objects. And these predictions for different frames will be the kinematics two different frames would describe for the same events.
But Maxwell in vector form didn't tell us how the electric and magnetic fields transform between frames. We know the equations should be the same, but we don't know how the fields, the solutions, should change. So you should take it backwards and say that the fields should transform in a way that gives us the same dynamics. And that happens because we define the fields in terms of forces. So we need the Lorentz Force law, for example, in order to know what the fields should be in the different frames in the vector form.
So we have to assert how they transform to make it so they give the same dynamics for motions of stuff when we combine with Lorentz Force and some laws of motion.