I recently had to ask a professor for a recommendation letter, and let me tell you, it can feel super daunting! I kept my email straightforward: I included a reminder of the class I took with them, explained what program I was applying to, gave a clear deadline, and I even attached a one-page brag sheet highlighting my relevant experiences.
One key tip that made the whole thing feel less awkward was making it easy for them to decline if they were busy. Just a simple line like, 'If you're unable to help, I completely understand!' goes a long way. This way, there's no pressure on your end!