How does cancer metastasize?

Enters circulation. The biology behind it is complicated and has been studied for decades by cancer researchers. However in a nutshell, the tumor gains access to blood vessels and lymphatics where it can then travel to distant sites such as lymph nodes, liver, lung, brain, and bones most commonly.
Local, blood, lymph. Cancer spreads three ways: by directly invading surrounding organs and tissues, or through the lymph nodes and blood vessels to distant organs.