Besides echoing what's already been shared, I also keep a close watch on the language I use toward myself, taking care to be kind and gentle in my dealings with myself. Instead of the old "I can't believe I just ________. I'm so stupid!", I might replace this with "Oh, I made a mistake. I guess I'm human after all!"
For me, the essence of "Let It Begin With Me" is that I first treat myself the way I'd like to treat/be treated by others. Only when I'm first treating myself in a compassionate, loving manner can I truly treat others in the same way.