What is the underlying explanation behind fictitious/pseudo forces?

Question

What is the underlying explanation behind [fictitious/pseudo forces](https://en.wikipedia.org/wiki/Fictitious_force)? 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

Answer 1

There are WAY too many words here....

Here's the short answer: F = m*a.

The sequence of events:

1. you and the bus are going the same (constant) speed
2. The bus starts to slow down. With no friction or other mechanical restraint, you will keep going (Newton's first law). If there is a mechanical restraint, then the bus applies a force to you to make you decelerate at the same rate the bus is

An interesting side-note: If the passengers are not restrained, then the braking force required to slow the bus is less---until everyone hits the windshield....