Why do the principles of physics get more complex when the frame of reference undergoes acceleration?

Your question is "why is X true," but I don't think X is necessarily true.

If you describe the motion of a projectile in the earth's frame, A, it's pretty simple. If you describe it in a free-falling frame, B, it's also pretty simple. It's not really obvious which of these is the inertial frame. A relativist would actually consider B to be inertial. In one frame there's a gravitational force. In the other there's none.

It's not even true that the laws of physics have to be expressed in a frame of reference. The standard way of expressing general relativity doesn't even involve a frame of reference.

I will address your question in terms of the motion of an object from a classical mechanics point of view. Newton's laws were developed to explain the observed motion of an object. Consider the second law $vec F = m vec a$; an object changes velocity (accelerates) only when subject to a force. A complicated force results in a complicated motion; for example, a force in three directions results in acceleration in three directions.

Viewed from a non-inertial (accelerating reference frame), an object appears to be accelerating due to the acceleration of the reference frame. Since the acceleration of the frame can be complex (translational and rotational) the inertial (fictitious) forces needed to account for the motion of the object viewed from that frame can be complicated.

I disagree with user284297. Things do get more "complicated".

If I understood correctly, you mean "complicated" in the sense of the need to describe the relative motion of bodies with, say, extra (inertial) forces. This happens because, in the inertial-to-accelerated scenario, you have to take into account certain things you would normally ignore if changing from an inertial frame to another, such as centripetal forces and so on. Then, why is that so?

I believe a simple answer to a such question is that the laws of physics (as we know it) are built so that all physics experiments done in inertial frames are equivalent to one another. In essence, inertial frames of reference are classes of equivalence such that the physics is described in a similar way. Moreover, the fact that the laws of physics essentially depend on the relative position of each body of a system (translational invariance) plays a big role here, since then you can translate your experimental apparatus somewhere else (to another inertial frame of reference) with similar (controlled) external conditions and your experiment will give the same result.

In addition, the notion of "inertial frame of reference" is a relative concept. That is to say, it is inertial with respect to a given frame (usually, the frame of a lab or even from the perspective of someone/something). As an illustration: two accelerating frames of reference can be inertial to one another, they only need to accelerate in the same manner. Between co-accelerating frames, both will feel the same inertial forces. Thus, one is essentially as "complicated" as the other.

Now, the way in which you define this inertial class equivalence will change drastically what is more or equally "complicated". For instance, Newtonian and Einsteinian gravity have very different notions of inertial frames. For Einstein, all bodies subjected only to gravitational effects are inertially equivalent. For Newton, objects in free fall are subjected to the force of gravity.

In summary, you have the same "force formulas" to describe the physics happening in all inertial frames. So, physics is described in a similar way when changing from one inertial frame of reference to another. This notion can be laid down in a more mathematically rigorous way when talking about the symmetries of a system. But it would not be very enlightening to delve into rigorous math at an early stage.

Actually the principles of physics remain the same in all reference frames. That is one of the two foundational postulates of relativity. What seems to change is the measurements made depending on the reference frame. In an accelerated frame of reference, things that were invariant in an inertial frame such as force, inertia and change in energy also become frame dependent. So there are more variables to work with, and also things such as time dilation and length contraction come into play, and so the complexity increases