When you unscrew the circuit board and take it out, you can see that the circuit board is a thin piece of fiber glass that has thin copper "wires" etched onto its surface. Electronic parts are assembled on printed circuit boards because they are easy to mass produce and assemble. In the same way that it is relatively inexpensive to print ink onto a sheet of paper, it is inexensive to "print" copper wires onto a sheet of fiber glass. It is also easy to have a machine drop the parts (the chips, transistors, etc.) onto the sheet of fiberglass and then solder them on to connect them to the copper wires.
When you look at the board, you can see a set of contact points for the buttons. The buttons themselves are made of a thin rubbery sheet. For each button there is a black conductive disk. When the disk touches the contacts on the printed circuit board, it connects them and the chip can sense that connection.
At the end of the circuit board there is an infrared LED, or Light Emitting Diode. You can think of an LED as a small light bulb. Many LEDs produce visible light, but a remote's LED produces infrared light that is invisible to the human eye. It is not invisible to all eyes, however. For example, if you have a camcorder it can see the infrared light. Point your remote at the camera and push a button. You will be able to see the infrared light flashing in the viewfinder. The receptor in the TV is able to see infrared light as well.
So the basic operation of the remote goes like this: You press a button. When you do that you complete a specific connection. The chip senses that connection and knows what button you pressed. It produces a morse-code-line signal specific to that button. The transistors amplify the signal and send them to the LED, which translates the signal into infrared light. The sensor in the TV can see the infrared light and "seeing" the signal reacts appropriately.