How to connect a Printer WIFI module wires to an Arduino?
"Simple" is extremely hard these days. What you are looking at is a small computer in itself; the Broadcom chip (datasheet) contains an ARM processor and is an entire computer in itself to run the WiFi. The datasheet says that it speaks either SDIO or USB. The square pad on the board usually indicates pin 1, although that doesn't tell you what pin 1 means.

I would take a look at and probe voltages on the other end of the connection as well. I suspect that the black and red paired wires are power and ground, but we don't know what voltage. Either 3.3V or 5V are likely possibilities. Maybe one of each. Get it wrong and you'll destroy the board.

This page: http://wikidevi.com/wiki/Foxconn_U98H035 and the linked forum post confirm that it's USB, although infuriatingly neither page properly describe the pinout. It's basically identical to a cheap wifi USB dongle. Which is very little use with an Arduino.

Its approximate value is $25 as a spare from HP or $10 on ebay.