Slave-making occurs in two ways. A queen ant may go to a nest of a similar ant species and kill the resident queen. She takes over the nest and uses the workers to bring up her own eggs. The original species of ant will eventually die out without their queen to reproduce for them.
The other way in which the slave-making occurs is that workers from a slave-making ant colony will steal other nests of their larvae and pupae and raise them as slave workers.
A video explaining this behaviour can be seen on the honey pot ant page