Proteinase K is important in order to degrade proteins that could affect the quality of DNA preparations for subsequent reactions, for instance restriction and modifycation enzyme reaction that will be otherwise inhibited.
The major goal is to get rid of any enzymes, specifically DNAses. Proteinase K will chew up the DNAses before they can touch your DNA, leaving it whole and intact.
Proteases catalyze the breakdown of contaminating proteins present in the solution to its component amino acids. It also degrades any nucleases and/or enzymes that may be present in the sample. This is of vital importance since these chemical compounds can attack and destroy the nucleic acids in your sample.
Proteinase K is an enzyme that cleaves the peptide bond in proteins next to the carboxyl group of hydrophobic amino acid residues (aliphatic and aromatic). Proteinase K itself is a protein, but it is resistant to denaturation by heat, detergents, and chaotropic salts and will continue to function happily in them as long as the temperature/concentration is not too high. It is stable (and functional) up to 65 degrees C and will function in temperatures down to 25 degrees C (room temperature). It is often used at higher temperatures (50-65 degrees C) because most nucleases that would chew up your DNA are denatured/inactivated at these temperatures.