Answering specifically regarding USB-C connectors (not breakout boards in general):
USB-C is an extremely high-performance connector, able to transfer up to 100W of power and data at 5 Gbit/s (USB 3.0) or 10 Gbit/s (USB 3.1). As such, signal and power integrity of of utmost performance if you want the full capabilities of USB-C. A breakout board such as the one you linked appears to have neither: proper layout would involve impedance controlled stackup, well-routed diff pairs, and large power traces.
If you are only using the USB 2.0 capabilities of USB-C, requirements are much less stringent, but as Ignacio has stated, the linked board is inappropriate because it has no way to be mechanically linked to another board (i.e. do not rely on soldered connections for mechanical strength). Find a breakout board with screw holes, mount it to your other board (or nearby in the chassis), and solder wires between the boards.
Again, if you're developing a device that uses the full capabilities of USB-C, the fact that you're asking about how to use breakout boards demonstrates that you have a huge amount of things to learn. 10 Gbit/s is no joke.