Why does the CPU consume more battery during computationally expensive operations?
There are two effects that influence the amount of power a chip needs. The static power (transistor leakage) and dynamic power. The static power depends on the technology used to build the chip. Therefore even at 0Hz clock the chip draws current. The dynamic power is what you're interested in.

The reason for consuming more power is the number of transistors switching per unit of time. The more transistors switch on or off per second the more power is needed. 

Some CPUs indeed throttle their clocks going into different modes. That reduces the number of switches. Others have a capability of turning parts of the CPU off to save energy.

The power heavily depends on which parts of the CPU (die) are used at a given time.