Think of it like this: The current is divided.
Imagine the button as a small resistance when it's pressed, and a large resistance when it's not. If the button resistance is smaller than R2, more of the total current in the branch will flow through it. If the button resistance is much larger than R2, less of the total current in the branch will flow through it.
Now imagine the resistance of the button approaching zero. As it gets smaller and smaller, it takes more and more of the total branch current, until there's essentially none flowing through R2.
Now imagine the resistance of the button approaching infinity. As it gets larger and larger, it takes less and less of the total branch current, until there's none flowing through the switch.
When you understand that a closed switch is close to zero ohms (but never zero, unless we're talking about superconductors) and an open switch is essentially an infinite resistor, you'll understand why most of the branch current flows through the switch when it is closed, why no current flows through it when it's open, and why there's always a tiny trickle of current through R2.
An ideal switch can be approximated as a zero ohm resistor, but in practice there's always some small resistance with a closed switch. An open switch is easier to envision as a infinite resistor, so no current flows through it when open (air is a pretty efficient insulator, unless the voltage gets too high...)