This is a jukugo word, which usually means on'yomi readings from the kanji. This is an exception, though. When you're counting days (at least one through ten), the kun'yomi readings are used. So let's think about that sixth day - what does it entail?
The cow (moo / むい) is standing on top of your car (か). Why? Because you only feed the cow once a week, and that is on the seventh day. He's standing there on the sixth day because he's super hungry and knows it's almost time.
Where did you two meet on the sixth of last month?
What were you doing when you took six days off of school?
A bear came into the bar on the sixth day of the month.