The bleeding is due to irritation and inflammation... and often, some cats will over-groom that area because having a UTI is uncomfortable and they seem to think grooming will soothe the pain... which oftentimes can lead to inflammation of the genitals (as you can imagine...).

One of the most common signs of a UTI besides frequent/straining to urinate is drops of blood in the litterbox... nearly every UTI will have blood in the urine sample, even if you can't see it with your naked eye it will show up on urine strips or after they spin the urine down and examine it under the microscope.

Glad to hear pills are going well!!!!! Hope the healing process is fast and the recheck urinalysis after the antibiotics shows no more infection!

EDIT: The blood you're seeing in the litterbox isn't frank blood but blood-tinged urine. It's probably less blood than you'd think (think if you cut your finger and put it in water - it would seem like there was a LOT more blood!!).