The LED-Wiz is a so-called "open collector" sink-only device. Its outputs will never "supply voltage". They essentially just turn on and off a path to ground. This is why you can use it with various voltage supplies to fit your application requirements.
You can use this to drive an SSR or similar, but you need access to the "low side" of the current loop. If I've found the board I think you've said you have, it requires a current SOURCE as it has it's own sink (an NPN transistor). A bare SSR can be driven just like you would drive an LED (read the LED-Wiz docs - note that most SSRs have built in resistors), then the output side of the SSR can switch anything its rated for. The LED-Wiz can also drive a mechanical relay, too.
You may be able to get by with using a "strong" pull-up resistor to provide the pilot to the NPN on your SSR board and letting the LED-Wiz "ground it out" to turn things off. This would result in an inversion but may be usable. It's not a simple no-passives direct connection, though.
In a nutshell, your Arduino SSR board has circuitry that is undesirable in this application.