She was walking by me.
This refers to the process of her walking past; it refers to the moment when the action occured; it doesn't specify whether she actually walked past. It is likely to be followed by an explanation of what happened during her walking past - what interupted that moment.
She was walking by me, when a car came speeding round the corner and almost knocked her down.
She walked by me.
This indicates a complete action.
She walked past.
I am not a teacher.
Student or Learner