There are many answers to the question why is the Earth bulged at the equator, see e.g. here, but almost all of them involve centrifugal force. Since it's a fictitious force, how to we explain this effect in an inertial frame. I guess there would be a similar answer to when we rotate a stone tied to a string, as we rotate it faster, the stone moves away from the axis of rotation. Please answer why does this happen, and don't involve the use of centrifugal force.
The stone (single particle)
The so-called centrifugal force is indeed a fictitious non-existing force in the inertial frame.
But the centrifugal effect is very real. That effect is causing the bulging. The counter-intuitive thing is just that it is not a centrifugal force which is causing the centrifugal effect.
Rather, the cause is inertia. Think of your stone-in-a-string example - for instance a game of tetherball.
- Throw the stone and it will continue straight ahead. This means that it will move away from you and from the tetherball pole that it is tied to.
- The string holds back in it so it doesn't move away. In fact the string pulls in it from the side causing a sideways speed component - causing the stones path to tilt inwards towards the pole.
- In this new position the stone still, due to the momentum carried via its inertia that it still possesses, wants to move straight. The path is slightly tilted from before but still it tries to just continue straight.
- Again the string pulls sideways, and again the stone turns.
- The string will do this pull at every moment and always sideways, so the stone will turn constantly.
And this is how the string causes a circular motion. The stone will at every point have a tendency to move outwards, away from the circle. Not because a force pushes it inwards but simply because it with its inertia "wants" to continue straight from every position. "Continuing straight ahead" corresponds to moving away from the circular path.
So, a centrifugal force may feel like a convenient intuitive explanation. But there is no such force. It is just an illusion, a "feeling" when we are sitting in the car and are squeezed towards the side. In fact it is not us who are being pushed outwards; rather it is the car which is being pulled inwards (due to friction). It is not your body which is moving out into the car door, it is the car which is moving inwards into your body.
So it is in fact directly opposite - there's no centrifugal force outwards but instead there is a force inwards which causes the inwards acceleration $a_c$ that causes the turning. We call that inwards force centripetal.
The inwards centripetal force that causes this centripetal acceleration (which causes turning) has to be larger if the speed of the object is larger, since a higher speed "makes it harder" to turn it, so to speak.
Because of this inwards force the object is constantly turning. But it doesn't "want" to turn, it wants to continue straight, and this is what gives rise to the feeling of a centrifugal effect - the tendency to move away from the circle.
The planet (continuous body of particles)
Now extend this idea to every particle on the planet.
The planet rotates about its axis. Those particles that are farthest away from the rotation axis move faster (in order to make it around in the same time as those particles that are closer and thus have a smaller circular path). Thus from the above equation those particles must experience a larger centripetal acceleration in order to turn properly. Such a larger centripetal acceleration requires a larger centripetal force.
And there we have it. The centrifugal effect is larger where the planet is fatter - so that would be at the Equator. Thus it is bulging out in these areas.
If you're standing stationary on a body rotating with a constant angular velocity, the total force on you from gravity and the ground has to equal the centripetal (not centrifugal) force associated with your circular path.
Suppose you're on a spherically symmetric rotating planet. At the poles, the normal and gravitational forces are equal and opposite, and their sum is the centripetal force (zero). At the equator, the gravitational force is the same as at the poles, and the normal and gravitational forces still add to the centripetal force, but the centripetal force is nonzero and downward, so the normal force is smaller than at the pole (you feel lighter). At any other latitude, the centripetal force isn't radial, so the normal force and gravity alone can't keep you fixed on the surface. The only other available force is friction, which must point toward the pole.
If there isn't enough friction, the net force on you will not be the centripetal force but the centripetal force plus a tangential force pointing toward the equator. The surface of the planet itself is subject to the same force, and (if it's like Earth) it isn't rigid enough to avoid flowing over geological time scales. This tilts the ground (which tilts the normal force) and also redistributes the planet's gravitational mass (which tilts the gravitational force). At equilibrium, the normal force and gravitational force (which no longer point along the same line) add to the centripetal force everywhere on the planet, so you stay in place without friction.