How can I use castellated vias with a breadboard?
The connections on that board are called castellated vias.

To connect it to a carrier board, you place pads on the carrier that line up with the castellations and then connect the two with solder.

For prototyping, if the pins are spaced correctly, you can solder them directly onto the copper side of stripboard (Veroboard). Just remember to cut the strings underneath the daughterboard so you don't accidentally short out something on the back.



There is an example SparkFun tutorial here. In case of link rot, here are some pictures from that example (CC BY-SA 4.0).

An example daughter board:



An example carrier board:



And the two connected: