Gentlemen, the VIC-II
The largest silver box on the motherboard--the one with the multiple ventilation holes, houses a large purple chip called the VIC-II (for "Video Interface Chip"). This chip handles all of the C64's graphics duties. The silver enclosure provides RF shielding and acts as a heat sink for the inordinately hot-running VIC-II.
The VIC-II chip, which was originally ceramic, was changed to plastic in later runs of the C64. Unfortunately, the plastic chips didn't conduct heat as well as the ceramic chips, and the VIC-II became a common point of failure in many C64 machines. It's one of several instances where cost-cutting measures hurt the good design and longevity of the Commodore 64.