There is likely no deeper meaning to the “stirring up of the waters” other that the normally tranquil pool became noticeably disturbed by bubbles, waves, currents etc. The stirring would be necessary because the Angel of the Lord probably wasn’t visible. The stirring allowed for those waiting for healing to know that the Angel was present and they would try to be the first into the pool for healing.
As an aside, this verse doesn’t appear in most of the earliest New Testament manuscripts and in most new translations it usually isn’t translated. It was probably added later to try and explain to the reader the reference the sick man makes to the stirring of the waters in John 5:7.
For more information take a look at this sermon by Pastor Ray:
Have You Stopped Blaming Others For Your Problems
Answered by Mark Pritchard