Question
What causes the browning patches (St Augustine grass) just along the strip near the curb? What should be done to recover?

Answer
If the boundary of the brown strip is fairly straight and well defined, it could be chinch bugs. These insects start at the driest parts of the lawn so that in itself may indicate a shortcoming in irrigation coverage. If the edge shows curves, that is strong evidence that the irrigation heads are not covering as much as they should.
Soap flushing at random along the edge of the affected area will uncover insect activity and testing the irrigation system for adequate coverage will point up areas for corrective action.