@cabarney I started a blog post/tutorial/guide/project (that I really need to go back and finish) about using the servo module to control arbitrary actuators. For now it lives here on the projects portal. ...But like I said, it still needs some work.
The basic premise is that the servo module provides the control signal and that a separate controller interprets that signal and handles the implementation, which, as you noted, usually amounts to an enormous power boost of some kind. In the case of a hobby servo, the signal interpretation and power boost both happen inside the servo's casing, but bigger things typically use separate controllers and actuators. Here's an image (from that thing I wrote) of the what the setup looks like for a 60A motor controller powering a really big brushed DC motor:
Power comes in on the right side of the image from a PC power supply, the thing with a fan on it is the motor controller, and the big red button is an emergency stop (habits from working on large mobile robots die hard ). The rest should look familiar.
In @mrcwinn 's case, the servo module is commanding an H-bridge IC, but the module's output is incredibly versatile. It can be used to drive most hobby speed controllers (which allow for the control of arbitrary actuators), as well as LEDs, the gates of transistors, etc.
On to your questions!
Use diagram 2 on page 5 of the datasheet for reference.
INX2 control the two sides of the H-bridge and
ENABLEX turns the bridge on or off. You would connect
INX2 to pins on the servo module.
ENABLEX could probably work on either a servo pin (set to 0% or 100% duty cycle) or a Tessel GPIO.
This chip does some of the legwork for you (it controls the upper and lower halves of each side of the bridge such that there's no shoot-through current), which means that the logic signals on each pin are interpreted as "open the top and close the bottom" or vice versa for each side, for a total of four states per bridge.
You have a few options for how you control the bridge, then, but the simplest way to turn the motor in one direction would be to hold one side of the bridge high or low and set the duty cycle of the other side according to the speed you want. Alternatively, you could set the two sides of the bridge as needed for the direction and then PWM the enable line...but that's probably not as nice to the chip's internals.
You'll also want to increase the module's PWM frequency considerably (25kHz is about right) and call
configure(channelNumber, 0.0, 1.0) for each channel to open up the full PWM range. Note that 50% will then be stopped and 0% will be full speed in one direction, 100% full speed in the other.
Related: this is a great resource and has more than you ever wanted to know about H-bridges and motor control theory.
ENABLEX lines are connected to the servo module/Tessel as described above
V_CC is connected to Tessel's 3.3V rail
VS X is/are tied to the power source for your motors(s), probably either the positive side of a battery, PC power supply, or AC adapter
SENSEX lines are tied to GND or to a current measurement system of some kind (make sure that GND is common between your motor power and Tessel)
TSD ARM is wired to a GPIO on Tessel
A breadboard is is probably safe for the motor you linked, but I wouldn't trust it for anything much bigger (anything above ~30V and/or 5A is probably too big).
...all that said, if I were looking to drive motors with Tessel and had no parts on hand, I would buy a speed controller, not an H-bridge. Even though they cost more (less so for low power stuff), their interface is cleaner and the "fine grain control/headache-to-benefit ratio" is better in most cases.