In particle physics, for example, we add gauge fields, look for the covariant derivative and so on. All to find the LI form of the Lagrangian. Why do we need the LI form? My impression is that when quantities are LI we equalize conditions, but I don't see how.
Answer
Lorentz invariant quantities do not change from one inertial frame to another. For example, the rest mass of a particle is Lorentz invariant, and since it doesn't change from one inertial frame to another, and therefore is an intrinsic property associated with that particle.
Moreover, we require the Lagrangian (more precisely, the action) to be Lorentz invariant so that the Euler-Lagrange equation (both sides of which has equal number of uncontracted Lorentz indices and transform in the same manner under Lorentz transformation) derived for the fields be Lorentz covariant i.e., do not change in form from one inertial frame to another. Special relativity teaches us that the laws of physics should be covariant.
Edit: We replace ordinary partial derivatives in the Lagrangian by covariant derivatives to ensure gauge invariance (in gauge theories i.e., quantum field theories which are invariant under various local symmetries such as local $U(1)$ symmetry in QED, local $SU(3)$ symmetry in strong interactions and so on.). This has nothing to do with Lorentz invariance. Lorentz invariance is essential for any relativistic quantum field theory. Gauge invariance is required in gauge theories to ensure renormalizability.
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