What is the underlying explanation behind fictitious/pseudo forces?
The popular example of the bus: Lets say you are standing in a bus and the bus is moving with a constant velocity, we can therefore agree that you are in an inertial reference frame and therefore the law of inertia applies in your reference frame.
However, as soon the bus decelerates you feel a "push" forward, moreover the frame is no longer inertial, at this point we agree that the law of inertia doesn't hold for that frame since you changed the state of your motion while no force is acting on you in your frame. Consequently, in order to make up for the "discrepancy" between the law of inertia and such situations we introduce a pseudo force (stating that this is the force that caused us to change states) in order to be able to effectively use Newtonian mechanics in a broader domain.
This is the popular explanation as to why pseudo forces are introduced, however no one really touches on to the underlying principles of the occurrence of a pseudo force, so I'm looking forward for a more in depth explanation into the nature of a pseudo force (i.e why it occurs from a physical perspective?), rather than just saying that we introduce it in non-inertial reference frames, for reasons similar to the above.
If we assume its nothing more than just a human correction used for mathematical and physical analysis, and simultaneously we can't say that inertia is the reason we tend to fall forward in the situation stated above since it is a non-inertial frame, then what would be the explanation exactly to such tendency of changing states of motion in an example like the above
Best Answer
We actually don't introduce a pseudo-force as much as we introduce an acceleration. It is an acceleration which is experienced by all bodies in that non-inertial frame. From time to time, it can be convenient to think of it as a pseudo-force, but the deeper meaning you're looking for deals with accelerations, not forces.
In your bus example, when the bus starts decelerating, every object acquires an acceleration which corresponds to the effect of the reference frame decelerating. Thus, your human on the bus will accelerate forward unless a force generates an opposing acceleration.
A more interesting case is a rotating frame. A rotating frame is non-inertial, and the equations of motion within that frame include a centrifugal acceleration $a=\frac{v^2}{r}$ away from the center of the rotating frame. If no force pushes on the object, it will accelerate away from the center at that rate. However, in most interesting rotating frame problems, there is a force in the opposite direction as well. In the case of an orbiting body like the ISS, that force is the force of gravity, $F=mg$ towards the center of our rotating frame. This generates an acceleration of $g$, and when the acceleration $g$ from the sum of the forces is equal but opposite of the acceleration from the non-inertial reference frame $\frac{v^2}{r}$ the object appears not to move (in the rotating reference frame).
Likewise, if you are spinning a weight on the end of a string, it's the force of tension on the string which directly opposes the accelerations from the non-inertial reference frame.
The idea of a pseudo-force comes about when it is not intuitive to think about these accelerations. Consider the case where you're on a gravitron, which is the carnival ride that spins really fast and pins everyone up against the wall. In this case, it is not intuitive to think about the difference between the accelerations from your reference frame and accelerations caused by the force of the walls pressed up against your back. Every part of your body feels as though there is a force pushing you outward. In fact, if you run the math, the effect of this "centrifugal force" pushing you outward is identical to the effect of an acceleration caused by the non-inertial frame multiplied by your mass.
This is where the pseudo-force comes from. At a deeper level, its really more meaningful to treat it as an acceleration, but in practice it can be convenient to model this acceleration as a force by multiplying the acceleration by the mass of the object. When we choose to deal with these non-inertial effects as forces, we call them pseudo-forces. In particular, we like to do this when we want to say the sum of the forces on a body (that isn't accelerating) is 0. It's convenient to think in all forces instead of having to mix and match forces and accelerations. It's also convenient to think this way because the intuitive wiring in our brains is typically built to assume inertial frames (even when that isn't actually accurate). But the "meaningful" math behind them is all accelerations.