However, how does a person outside the car rationalize this phenomenon
When the car initially accelerates, the person outside the non inertial frame of the car sees the pendulum bob as "staying in place" while the car accelerates forward, i.e., the bob does not accelerate with the car. The person explains it as due to the inertia of the bob as a consequence of Newton's first law, namely, that a body at rest tends to stay at rest unless acted upon by a net external force.
The person inside the car can only explain the initial backward acceleration, based on Newton's second law, as being due to some force acting on the bob, the same force that pushes the person back against the seat when accelerating. Since there is no contact force acting on the bob or pushing the person back against the seat, it is considered a pseudo or fictitious force.
Once the car accelerates and maintains its acceleration, the bob will stay in place in some backward position and actually experience a real acceleration along with the car due to the tension in the bob string.
"When the car initially accelerates, the person outside the non
inertial frame of the car sees the pendulum bob as "staying in place"
while the car accelerates forward, i.e., the bob does not accelerate
with the car. " Doesn't the air molecules inside the car bouncing
around have any effect on the motion of the Bob? I mean would the
result be same even if we did this in a space ship in vacuum?
If the bob had motion due to the air, that would mean it is being influenced by contact forces with the air molecules. Remember, Newton's 1st law states that an object at rest tends to stay at rest unless acted upon by an external force. If motion of the bob were due to contact with air molecules, that would constitute such an external force. Naturally, in the vacuum of space no contact with air molecules would be involved.
I think the confusion I'm having is the part about the pendulum
staying in place even when the car starts moving. It is like we can
clearly see the angle which the pendulum rope makes with the vertical
axis changing. So, clearly, the configuration of the system is
changing yet we say that the pendulum is staying in place
The pendulum bob only initially stays in place horizontally in the frame of the road due to no external horizontal force. But as the horizontal component of the string tension increases the bob moves with the car.
The diagrams below may help to visualize what's happening. The first digram shows the pendulum attached to the roof of the car from the perspective of an observer in the inertial frame (the road). We will assume no air drag force will arise. So initially there are no external horizontal forces acting on the bob.
Figures A-C of the second diagram show the progression of motion of the car/pendulum as seen by an observer on the road (the non accelerating frame). The acceleration begins at Fig A.
In Fig B the car/pendulum base has moved in the x-direction. The bob tends to stay in place horizontally due to its inertia with only a slight displacement (less than the car/base) in the x-direction due to a small horizontal component of the tension in the string.
In Fig C the bob is increasingly being pulled by the string in the x-direction due to the force in the string. Eventually the pendulum angle will reach a maximum value, depending on the car's acceleration, and the horizontal velocity of the bob will be the same as the car.
Hope this helps.