Variable Importance Random Forest on R
What the function is doing for each variable?


Record the Out-Of-Bag (OOB) accuracy for each tree.
"shuffle" or permute the values of that variable.  This means you take all the values of that variable in the data, and assign those values randomly back out to the observations, which is a way of introducing noise and getting rid of the signal that that variable provided.
Now it finds the OOB accuracy again, but this time the values for that variable are incorrect since we permuted it.  By introducing noise where your model expects signal, you should see a decrease in performance.
Compare the original accuracy in (1) to the accuracy in (3) for each variable.  If the model performance decreases a lot for a variable in step (3) compared to (1), then it is deemed to have greater importance.


Why does removing the most important variable not have a negative affect on accuracy? (my guess)

Probably because that important variable is correlated with some other variable(s) you have.  Your model can capture the information contained in that missing important variable by using a few other variables to make up for it.  When you drop the important variable, which other variables see a notable gain?