If the integrated circuit die is very small what is the role of the extra circuit packaging?
What is the purpose of the packaging?


Protecting the IC against light (light will induce current flow in a PN junction)
Protecting the IC against moisture
Together with the leadframe take the connections of the IC further apart. These can be as closely spaced as 100 um which is too close for standard cheap PCB manufacturing. The leadframe + package expands this to something more usable like 0.4 mm up to 2.54 mm (DIP/DIL packages)
Make the IC easier to handle by humans. A  DIP package can easily be used and exchanged in a breadboard or in a socket.


And what are the materials that IC packages are made of

The leadframe: tinned copper or metal so that it can be soldered easily

The black part: usually molded plastic, sometimes a ceramic material.

Some ICs can be bought in a CSP (Chip Scaled Package) which actually means no real package at all, on top of the chip a redistribution layer is made (which spaces the connections to what the PCB can use) and the IC is then mounted directly on the PCB. This technique is also called "flip chip".