General example:

curl -H "Content-Type: application/json" -X POST -u USER:PASSWORD -d '{ "text": "FUNKRUFTEXT", "callSignNames": ["RUFZEICHEN"], "transmitterGroupNames": ["SENDERGRUPPENNAME"], "emergency": false }' URL/calls

Example with real user data (login credentials not working; they have to be replaced by the real ones):

curl -H "Content-Type: application/json" -X POST -u dl1acb:sehrgeheimespasswort -d '{ "text": "DL1ABC: Das ist eine Sendung über die REST-API", "callSignNames": ["dh1xyz"], "transmitterGroupNames": ["dl-bw"], "emergency": false }' http://dapnet.afu.rwth-aachen.de/api/calls

For a python example, have a look on the implementation from Philipp DL7FL: https://github.com/DL7FL/DAPNET/blob/master/DAPNET/senden.py