Heavy Sleeper is the 3rd cutscene in the game, 2nd if you are not counting Batteries. Even though this is not a major cutscene, this cutscene is still quite influential, since this cutscene unlocks the Lizard, and will then be usable after this.
The Skater has just finished Level 40, and runs into the sleeping Lizard. He starts to apologize to the Lizard, saying he was caught off guard when the gravity changed. Then he realizes that the Lizard is fast asleep, and moves on. The next slide displays a line of text saying "Choose your character," and gives the option of picking either the Lizard or the Skater. Clicking the Skater will allow you to continue onward in the Main Tunnel as him, whereas choosing the Lizard extends the cutscene and lets you begin to explore the newly discovered B-Tunnel.
The extended cutscene fades to black after picking the Lizard, then shows the Lizard waking up after some time. The Lizard wanders back through the previous level into the B-Tunnel. The screen will then cut straight to the Lizard playing Level B-1, the first level of the B-Tunnel.
Heavy Sleeper is unique in that it is one of the two cutscenes to feature a 'Choose your character' screen, with the other being Coming Through. It is also the only one that has the possibility of an extended cutscene. Viewing Heavy Sleeper in the cutscene gallery will show the full, extended cutscene every time. However, the 'Choose your character' screen is absent from this version, similar to Coming Through.
This cutscene is only activated if you play through and finish the level as the Skater; every other character, including the Lizard, will just progress straight to Level 41 with no interruption.