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.
Best Answer
Real forces satisfy two fundamental requisites
They do not depend on the reference frame
They satisfy the third law of classical dynamics
Pseudoforces violate both requirements. Indeed, they appear only in some reference frames called, non-inertial in addition to real forces and they are added just to impose the validity of $\vec{F}=m\vec{a}$ also in those reference frames. Secondly, all real forces always arise in pairs in the Newtonian formulation of mechanics: the body B exerts a force $\vec{F}$ on the body A and simultaneously the body A exerts the force $-\vec{F}$ on the body B. If $\vec{F}$ is a pseudoforce acting on $A$, there is no $B$ and there is no $-\vec{F}$ acting on it.
(I stress that the notion of pseudoforce is proper of Newton's formulation of classical mechanics. When passing to more general relativistic formulations, that distinction between forces and pseudoforces is not so sharp also because the language is different, and pseudoforces share the same geometric nature of the gravitational interaction: they are no longer "forces". So that it is difficult to directly compare the two scenarios.)