I think she's trying to say that she was walking down the street with ease and confidence. The way someone walks when they're in a very familiar place, and have walked there often.

Strutting is walking in a proud manner, almost walking to show off your appearance. As an example, when one of my friends is looking especially good i may encourage them to " strut their stuff" I think tutting is more of a sound word in this context. It is the sound of her shoes hitting the pavement as she walks evenly and with pride ( as she struts!). Don't worry about this word too much though because it is never used in conversation. I think she just used it because it sounds good with the word strut.