It just means 'later in the day' or 'later that day'. I don't think the 'on' serves any real purpose.

It means that during the same day, something else happened later. So they studied the map maybe 4 or more hours after doing some other action that day. I am trying to think of a context in which this particular sentence would happen. Hmmm. Okay, say two people visit Montreal and they don't have a map. At 9AM they go to a used bookstore and buy a map. They then head back to the hotel to take a nap. Four hours later, they take the map out and study it carefully. They study it carefully because they want to go do something interesting and they need to know where the day spas, local parks, nightclubs, and other points of interest in the city are located at. :) So "They bought a map at the book store but studied it carefully later on in the day."