Ugh, what a horrible data sheet.
You need different hardware - both TLC5940 and TPIC6C596 are current sinks.
You need something to source current.
I suggest cd74AC164 to drive the row pins.
http://www.digikey.com/product-detail/en/CD74AC164E/296-4235-5-ND/375630I don't read the display as being very bright, just 3-5mCd at 20mA, so TLC5940 might be overkill here.
Use two '164s on the Row pins with current limit resistors to source current, two 6C596s on the column pins to sink current.
Set up the anodes, turn on a cathode, hold for 1-2mS, turn off;
set up the anodes, turn on next cathode, hold, turn off;
continue for all 10 columns.
Just so you know - the display is not RGB - its all green, with 4 red and 3 yellow blocks.