I was super anxious about approaching my professor for a recommendation letter but realized I needed to make it easier on both of us. I sent an email that reminded them of the class I took, explained why I was applying to the program, and included a deadline.
To keep it casual, I attached a one-page brag sheet about my accomplishments. This way, they had all the info handy and could write a strong letter without having to sift through my resume. Also, I added a line saying it was totally fine if they were too busy to help. Giving them an easy out lessened the pressure, and it felt more natural!