Thanks for the info! The text on the control chip is incredibly tiny, and reading it reminded me that I'm due for an eye doctor appointment.
Anyway, the ID appears to be SDRX2BDS, and there is a string of numbers beneath it that reads 11120006. I've not yet Googled, but am about to do so.
You don't have to - that's the RX2 half of the TX2/RX2 chipset (if you look inside the controller, you'll probably find something labeled "SDTX2BDS" or similar). Now, go to that thread I mentioned, and read it. There's a link in that thread to the PDF of the spec sheet for the chip.
You're first issue is to determine what the power source is for the chip, and what it is outputing as a logic HIGH on it's control outputs. You'll need a multi-meter for that. Once you know both of those things, then you'll be set to go to the next step.
I see that the chip is on a "daughterboard" of a sort. That's both a good thing and a bad thing. Good, because it means you can trace with your multimeter and your eyes the leads from the chip to the pins/holes on the "main" board. Bad, because it makes checking the pins and such a bit awkward (especially those near the main board side). But once you have things traced out, you can then carefully remove the daughterboard, and solder wires in place where the pins went, then supply signals to those pins somewhat directly from the Arduino. You may need to add current limiting resistors in between (if they are on the daughterboard and not the main board; hopefully they are on the main board side). Likely power and ground for the daughterboard come from the main board, so those pins may make things really convenient.
But you first need to figure out those voltages. Likely it will be either 3 volts or 5 volts used for the power to the chip and logic HIGH; if the chip is running on 3 volts, then the logic high will likely be 3 volts or close to it. One way to make a "first guess" is to find out what kind of battery it uses; if it is a 6 volt or greater battery, likely it is running the chip at 5 volts (which makes things much easier to interface with the Arduino). Otherwise it is probably a 3 volt system, which means you'll want to do some level shifting on the outputs (you can do this with resistors or diodes, or you can purchase specialty 3.3V <=> 5V level shifters; Sparkfun, among other vendors, sells them).
If you keep the receiver daughterboard intact, and you label the pins on the edge in some manner - you can use that for other control projects (with the transmitter)...
Just take your time, and be very patient and careful, documenting everything along the way. Read that thread I mentioned; you'll note that the OP of that thread ended up messing up his first car, but then got the second working (though IIRC, that one died too) - so be aware of that possibility. This isn't a "fool proof" operation, but many people have done it.