This expression can be literally translated by “to have a fat morning”. However, it means to sleep in, something you might like to do once you are off! Faire la grasse matinée dates back to the 20th century. However, grasse matinée was common in the 16th century expression dormir la grasse matinée (“to sleep the fat morning”).
So why the adjective grasse (“fat”)? There are two explanations. First, the adjective comes from the Latin crassus, “thick”. It would refers to the idea of having a thicker or longer morning than usual. Second, gras evokes laziness and the soft feeling of staying in bed all morning. Ready for you next grasse matinée?