It's an interesting use of the phrasal verb "make out" (which has several meanings anyway).
The full phrase is "to make out X in the Y". This means that Y is obscuring X in some way, and so it's difficult to see X. But you can... just barely.
"I can make out a house in the distance."
"She finally made out a silhouette in the darkness."
"Soon, the captain made out a ship in the fog."
"As near as I can make out" suggests that it is difficult to get accurate information (something is obscuring it), but with some effort, it can be seen/understood.