Explanation of protocol in DNA extraction experiment
The DNA solution has a fixed volume (e.g., 0.1 mL). To a suitable test tube is added 0.2 mL of buffer-saturated phenol. After closing the test tube the mixture is mixed well, typically using a vortex mixer, but possibly by careful, repeated inversion. The aqueous phase and the organic phase are barely miscible. The tube is centrifuged to speed up the equilibration, and concentrate the denatured protein at the interface. After centrifugation the aqueous layer is carefully removed to a fresh test tube.

That is one organic extraction. This is repeated with a fresh aliquot of buffer-saturated phenol (another 0.2 mL). Then the aqueous phase is decanted again and extracted with 0.2 mL of CHCl3.

According to the recipe you provided, the buffer-saturated phenol was purchased from a vendor (Sigma-Aldrich Chemicals, of St. Louis, Missouri, USA), so you can look up the ingredients and composition on their online catalogue (no doubt).

The buffer recipe you provided is the proteinase K digestion buffer. It is typically performed at an elevated temperature.

This protocol is designed to isolate DNA from the nuclei of hair cells. Fully half of the contents of a eukaryotic nucleus are chromosomal proteins, by mass, so the proteinase K digestion step not only breaks open the hair cells, but also helps to remove all of the unwanted, contaminating cellular and nuclear proteins.