What does the eagle appear to be staring at?

Which eagle?

The Eagle on the Great Seal of the United States of America faces to its own right (the viewer's left). Since the design specifies that the olive branch is grasped by the eagle's (own) right talon, the eagle faces the olive branch (but is not exactly looking at it; as it's considerably below the eagle's line of sight).

To the extent the eagle is "staring at" anything visible in the seal, it's the Words "E Pluribus" on the scroll/banner held in its beak.

The Presidential Seal is very similar to the Great Seal, but which direction the eagle's head is turned has varied over the years. It's NOT true that the eagle faces the arrows in wartime and the olive branch in times of peace, though the Presidential Seal was revised in 1945 and the eagle's head was turned towards the olive branch. This is not specifically because of the end of World War II, but rather because no one is quite sure why the eagle ever faced the other way in the first place unless it was simply to distinguish the Presidential Seal from the Great Seal ... heraldric animals traditionally face their own right ("dexter"); facing to their left ("sinister") is unusual.