I also drew a rough sketch of how I believe the circuit is designed for the motor controller. F_ic and B_ic are forward and back on the IC, respectively, and M_f and M_b are motor forwards and backwards, respectively. I'm sure I violated all sorts of standards for circuit diagrams, so forgive me, but I think the general idea is there.
In general it looks OK, though you show the output to the motors from the base of the transistors, and it's not really the "standard diagram" of an h-bridge you normally see, but close enough I think. You might want to compare it to the reference diagram in the PDF of the datasheet (I also note not seeing any flyback diodes - which is crucial in an h-bridge design based on transistors - unless those are -not- transistors, and are mosfets instead, which have an in-built diode across the source-drain; this would explain some things)...
I noticed you drew on one pair of transistors the base resistors, but not the ones between the RX2 IC and the first set of transistors - were there no resistors?
Looking at your diagram, the transistors F(ic) and B(ic) are probably being used in a couple of ways: 1) as a means to switch -opposite pairs- of the transistors in the h-bridge on/off (to prevent shoot-thru - which is bad - causes lost smoke), and 2) probably as a way to boost the output current of the RX2 IC to drive the larger transistors in the h-bridge.
If you can figure out or read the part number of these "buffer" transistors, it would be best to know that information. You can then determine (from the datasheet for the transistor) whether you can use it with the Arduino, what kind of base resistor would be needed, etc. Knowing the part number and such for the main drive transistors would also be helpful (especially if they are mosfets and not bipolars, which I am starting to seriously suspect).
You might be able to drive those transistors via their base resistors (I'm also surprised here where two of the other base resistors went?); it would use four digital outputs, of course (you might want to use a buffer driver IC in between), and you would have to code things carefully so that they get switched on/off properly without leaving two of the transistors on the same side of the bridge "on" (shoot-thru). Port manipulation would be best here (with all pins on the same port), or you might be able to carefully craft the software so that, knowing the last state of the pins, it would know how to set the state properly to avoid the shoot-thru condition...