Here are my thoughts.
Did he use it to hide his writing and ideas?
Very unlikely it is hardly difficult to mirror it back!
Because he wrote left-handed?
Again, unlikely the advantage is minimal.
I believe he did it to slow his thinking and tap his inner creativity.
To encourage inner silence and stillness.
To increase levels of awareness and higher, finer concentration.
To ground himself in the present moment, harness new, fresh thinking and ideas.
Using mirror writing slows you down and the pauses create a calmer, clearer, meditative state that increases the quality of your thinking, and unlocks rigid reasoning.
Perhaps he read it back to himself in front of a mirror in order to see himself whilst reading not out of vanity; in order to test what he had written from a different perspective and further contemplate his writing.
Try writing the words in the image below in mirror and you will see that it takes concentration not to write some of the letters normally.
Contrast that with the speed of writing the same words forwards and you will notice that you do it so quickly that not much thought is given to what you are doing.
Personally, I think it fosters deeper, truer, contemplation perhaps da Vinci considered concepts then mirrored to develop 360 degrees thinking, radial thinking like mind mapping.
I have also flipped the Mind Map so that you can read it normally:
See also The Qualities of Leonardo da Vinci Mind Map
If you have any thoughts on why da Vinci used mirror writing you can add a comment below.
Something else has struck me regarding Leonardo and this time specifically about the Mona Lisa. There is a theory that the Mona Lisa is possibly, or at least contains elements of, a self-portrait.
Looking at the name Mona Lisa I wonder if it could perhaps be a simple anagram of Mon Alias which in French is My Alias? Perhaps the Mona Lisa represents the feminine side/quality of Leonardo.
Something more to consider!