A side channel attack is described here. I did not quite get , how the power measurements are used to get the decryption key , can any body explain it?
I am not familiar with your side channel attack "here", but it looks like what I have read about before in those that monitor the processor to determine calculations. "The trick is based on the fact that the “ground” electrical potential in many computers fluctuates according to the computation that is being performed by its processor—including the computations that take place when cryptographic software operates to decrypt data using a secret key." Researchers using signals from grounding the computer through touch.
"Traditional power analysis would measure power consumption at a frequency comparable to the CPU's clockrate (a few GHz), and is foiled by dampening emanations at these frequencies. Our attack extracts the key using much lower bandwidth (a few kHz to a few MHz, depending on settings and duration). Our attack is also more resilient to filtering and noise."  Look at "Q14: How do you extract the secret key bits?" in this article. They find the secret key bits one by one sequentially by teasing out processor signals and what they represent. They go through some technical steps to get their timing correct, not like they walked by a computer in a coffee shop and happened to crack it.