The vector is the belief state of the robot.

1. Initially, it believes that it can be anywhere, so the vector is [1/5, 1/5, 1/5, 1/5, 1/5]
2. When it senses the "red", its beliefs change to match that observation, so the vector becomes [1/9, 1/3, 1/3, 1/9, 1/9].
3. Then it moves to the right. It is sure he moved to the right, so if it believed with confidence 1/3 that it was in the 3rd cell, it will be 1/3 confident that it is now in the 4th cell. Doing the same procedure for each position of the vector, his belief vector becomes [1/9, 1/9, 1/3, 1/3, 1/9].

The idea is that the vector represents the belief state of the robot and the values are the certainty of the robot about whether it is in that cell.

Lets take and simpler example.

p = [0.5, 0.0, 0.5, 0.0]
assume that you always move correctly when issue the move command

before moving, you have 50% confidence that you are in the first block and 50% confidence that you are in the third block.

What happens if you move one block to the right (before any further measurement) ?
Well, if you always move correctly you will have 50% confidence that you are in the second block and 50% you are in the fourh block.

p = [0.0, 0.5, 0.0, 0.5]

Or you can take an extreme example where you know your location :

p = [0.0. 0.0, 1.0, 0.0]

if you move to the left, what is the new probability distribution ?

i guess it has something to do with the fact that the robot doesn't know where it is , all it knows that there are 5 doors in a certain order [1/9, 1/3, 1/3, 1/9, 1/9] initially and that it is in a red cell not a particular cell , but just a red one.

i took "shifting the probabilities" for granted :D , now that you mention this , i wanna know what it represents too :D

Search for question, "2nd video, why is the belief shifted at all?"

Among the answers, look at the one that starts with "OK, I think I'm getting it now."

It discusses shift and eliminates the confusion of advancing to being in front of a second door. Discussion there should help with deeper understanding.