The device you intend to control must run on DC power, with a voltage at least above the LDO's drop-out voltage. In principle this probably means no less than 6 volts. The maximum voltage is limited by the absolute maximum input voltage of the LDO and its power dissipation (the 5v draw should be 1 mA or less, which implies that there should be almost no limit based on dissipation) and the absolute maximum Vds of the main power MOSFET. The maximum current drawn by the target device is limited solely by the absolute maximum Ids of the main power MOSFET (the gate voltage should be sufficient to saturate it or keep it fully off, so there should be no Vds dissipation concerns). In practice, the expectation is that the vast majority of equipment that you'd use would be powered by either 5, 6, 9 or 12 volts DC. 5 volts is too low for the LDO, but it would likely still work, as the controller itself can work well below 3 volts (it's default fuse configuration sets the system clock to 1 MHz, where it will operate down to 1.8V), and its operating voltage is not terribly critical (as long as its "high" output is enough for the Vgs threshold of the N MOSFET).
As designed, it requires center-positive 2.11m barrel connectors for the power, though with design changes on the board this can be altered to fit your needs.
2Hookup and usage
If you're using a Raspberry Pi as your watchdog host, then connect the control header up to pins 6 (GND) and 7 (GPIO4). Run the watchdog script every 4-6 hours or so from cron. The script must be run as a user who has GPIO permission.
Plug the router's power plug into the jack on the board. The power light on the board should light up. Plug the output plug from the board into the router. The router should power up normally. You can test the hardware by copying the watchdog script and deleting the stuff in the middle that actually checks the IP addresses. Setting GPIO4 low for 2 seconds should cause the router to power off for 10 seconds and then power back up. Repeating that low pulse in less than an hour should result in nothing happening.