Hmm - it looks like there is virtually nothing (save maybe for one youtube video) on how to use this board with the robot. So...here's how I would tackle it:
First off - those two chips are very common; the first being the L293D, a common h-bridge motor driver IC. Not the best thing on the planet, but you should be able to make it work. So - google for a PDF of the datasheet; you'll need it. The other is a bog-standard 7805 voltage regulator - it will convert a higher voltage (7-12 volts DC usually) to a regulated 5 VDC (just like the regulator on the Arduino). With proper heatsinking, you can get about an amp of current from it, but this thing doesn't have proper heatsinking, so don't expect that. However, it is likely only there to power the L293D and nothing else.
J1 and J2 are obviously for power - one is probably to power the L293D, the other for the motors - or maybe one is an input and the other an output (feed-thru? or maybe one outputs 5 VDC? dunno).
So - gather up as many images as you can (including all the ones from the Amazon item listing - plus any posted by reviewers). What you really want to look at are the ones showing the assembled robot. Hopefully, they can offer some guidance.
Also - take some high-resolution photos of the board you have. If you have a flatbed scanner, that can help immensely - scan that sucker at a high resolution (mainly the bottom of the board).
Then - get a multimeter. Also load up a paint program (Windows paint should work ok - no need to go fancy with photoshop, though it can help too - so don't rule it or similar programs like the Gimp out!). Load up your images.
Take a close look at the images. Compare with your board (use a magnifying glass - ideally, one of those magnifying lamps with the light work great for this). Note on the images where a trace starts and where it goes. If you can flood-fill the trace with the paint program, or use filters (photoshop, gimp, etc) to bring out highlights of the traces, it can help you figure out where things are connected. Take copious notes along the way.
With no power to the board, use a multimeter set for a continuity check, or on the lowest resistance setting, and probe between the header pins and the pins of the L293D. Note what is what using the datasheet, and the header pin silk-screening - a pattern should become readily apparent. Do other verifications and checks using the Amazon images, to see what wires are connected where.
I suspect that some of the headers (likely the ones with the small trimmer pots?) are for the optical speed sensors on the motors. Other headers are for the motors themselves. Some of the headers may be for other sensors or modules.
I'm curious as to what the IC U4 is - you didn't mention that? But again, get the datasheet for it, and start probing. Do everything carefully and methodically, and take a ton of notes. Eventually, you should be able to figure it out.
Once you do, document everything clearly - then dump everything you have on to github or something, and post a link to the Amazon page (as many languages as you can), to any blogs (I found a couple where the people seemed lost too), plus here - maybe also post an instructable.
The goal being of course that your hard work is shared with everyone else with this robot. You won't get any money for this, of course, but people will be very grateful. You'd be surprised at how much this helps people, and how they appreciate it. I've put out things of this nature myself over a decade ago, and I still occasionally get people posting a comment to me on a website (or out of the blue in some other manner - always a surprise) thanking me for the help. So let that be your reward, and maybe your goal.
If you don't do it, nobody else will - so roll up them sleeves, and start reverse-engineering that board.