How does heat exchanger work?

Every single plate in a plate heat exchanger (PHE) is actually a hollow thin metallic shell. The space inside every plate (or the space between every two consecutive plate walls) works as "passage" for the fluids. So the fluids flow along the "plane of the plates".

For clarity, look at the following diagram:

Every vertical black line shows a wall of a plate. Thus, each red box (or a blue box) is the space inside a single "plate". Red and blue arrows are respectively the hot and cold fluid flow directions. This is the basic diagram of a co-current PHE, the counter-current diagram can be made similarly.

A few other variations in the design are shown below.

The plates are stacked because that is the only way to keep the hot and cold fluids in contact in this setup. Also, this is the most efficient way overall.

Thus, I think you can see intuitively too that this setup gives the highest contact surface area, making it the most effective setup, contact area wise, which leads to higher heat exchanger that other setups.