The phrase "busy day" means that someone has a lot of things to do or a lot of activities happening throughout the day, keeping them occupied and not having much free time. Full definition
Why not indulge yourself in one of our luxurious spa suites, and unwind at the end of busy day of sightseeing or meetings. (wikiaustralia.com)
I know that the volunteers get started hours and hours before the actual race begins, and I thought that it must have been a very busy day for you. (looplooks.net)
We're home with our partner after busy days at work. (mindbodygreen.com)