It has two TI ADS1222 2 channel Serial ADC converters, for four sensors total.
The two smaller ics labeled TI 93 DBA3/DBAQ are probably opamps (can't find the part) but they should trace out to the ADC converters. Most likely, LMV932s, dual low voltage high quality opamps, but they don't match the suggested topcodes.
The four pin connections are Strain gauge wheatstone bridge connections.
The topmost (White) connection for each of the four bridge connections is the ground; relative to this, the two middle connections should be at 1.2 VDC, and the bottom should be at 2.7 VDC
The entire board acts like a standard wii remote, with a standard i2c wii balance board extension/accessory. It can be accessed via Bluetooth, or you could maybe tap into the i2c lines as well.
See http://wiibrew.org/wiki/Wii_Balance_Board for the bluetooth/hid descriptions and information.
Frankly, the bluetooth way is the easiest. Unfortunately, it's not as easy as the Wii Nunchuck being a simple i2c device with a connector.