What are the protrusions on the sides of the fuselage of the F-16I?
They are Conformal Fuel Tanks.

See http://israeli-weapons.com/weapons/aircraft/f-16i/F-16I.html:


  Conformal Fuel Tanks (CFT) - These tanks are manufactured by the  "Israel Aircraft Industries"  and increase the aircraft's amount of internal fuel by 50%. Their purpose is to significantly prolong the aircraft's flight range and ability to remain in the air. The tanks are installed the plane's back and conform to it's figure.


Israel does not have any heavy bombers, so they need the fighters to be capable of striking deep in enemy territory and since they can't be accompanied by tankers there, they need additional fuel.

As jwenting mentioned in comment, unlike most conformal tanks that are removable on the ground, these are fitted permanently.