By combining them, `打(engage in) + 交(exchange) + 道(the way/path)`, you get the vivid meaning of “engaging in the path of mutual exchange.” It's the art of navigating the pathways of human interaction.
`打交道` is an extremely common term used in a wide variety of contexts. The structure is almost always `跟 (gēn) / 和 (hé) + Person/Organization + 打交道`.