When it comes down to it, your basic mechanical pencil really is a remarkably simple device.
A plunger is used to depress the entire inner sleeve. A spring keeps the plunger under tension and draws it back after it has been pushed. The inner sleeve contains a lead holder with the end split into two 'pincers' that are held together by a collar. This collar is moved by depressing the plunger, which pushes the inner sleeve down while keeping the collar in the same place.
Perhaps the most critical part of the apparatus is the 'exit tube', located on the outer sleeve. This tube is the last thing the lead comes into contact with before being used, and functions as a way to retain the lead as it is pushed out by the pincers. Because the tube is functionally the same diameter as the lead, they are in complete contact and therefore exert a normal force on each other. Because neither of them are frictionless, this normal force combines with their coefficient of friction and forms a frictional force. It is this frictional force that holds the lead in place when one presses the plunger and opens the pincers.
Incidentally, the work one on the plunger by the operator is converted into both spring potential energy that is released when the plunger is released, and heat due to the sliding of the lead through the 'exit tube.' As such, Al Gore has suggested that we all "be kind to the great mother earth, and stop global warming by using fewer mechanical pencils." He always was an odd duck...