The previous answer may be a little misleading, as to give you a more detailed answer we need to know which MacBook you have. The procedure for removal of the heat-sink is different depending if you have an early/mid 2009 (pre-unibody) or the late 2009 A1342 unibody model.
With the early/mid 2009 (A1181) models you gain access on removal of the top casing, and the heat-sink is located on the upper side of the logic board - not exactly an easy task to remove the heat-sink to renew the thermal paste, but at least one doesn't have to remove the logic board. Not so on the unibody model - although access is simple by removal of the bottom casing, the logic board must be removed to gain access to the heat-sink on the upper side of the board.
I've worked on both models many times and far prefer the late 2009 A1342 unibody model, even removal of the logic board is relatively straight forward by following iFixit instructions.
One last point. It's always good practise to renew the thermal paste after several years use as it goes hard and flakey and becomes less effective at conducting heat away from the CPU and onto the heat sink. But my experience shows that if one doesn't have relatively easy access to the heat-sink for repasting, it's often easier to remove the fan for thorough cleaning and removal of dust/debris which results in a cooler running computer.
Incidently the photo in the previous post appears to show the logic board from the late 2009 A1342 unibody model. As it's been removed one can see the upper side, and the 'black thing' that was referred to is in fact the heat-sink.