It uses a deterministic algorithm to derive the public key from an initial value, which happens to be the same as the public key derived from a private key which has been derived from a private initial value.
In ELI5-Terms it's like two mazes on top of each other. Both mazes have the same layout, so if you know a path in the bottom maze you also know the path in the top maze. In the top maze the secret keys are hidden in various locations, while in the bottom maze the corresponding public keys are stored at the same locations. Only the maze on the bottom can be entered from the public, while the maze on top can be entered only via a guarded door. However at every place, where a secret key is stored in the top maze, there is a hatch in the floor which allows you to look down at the public key at the same place in the bottom maze.
In case it's not clear already: In this example a maze would be the deterministic algorithm, the entry points of the mazes would be the two initial values and the path through the maze are the parameters which make the deterministic algorithm derive a specific key.
EDIT: Of course, in the above maze-analogy the obvious catch of this setup becomes visible, too: As the bottom maze's door is open to the public, it can be entered by anyone. So a malicious maze visitor could go to the location of a public key in the maze and replace it with his own.
A more technological explanation can be found here:
https://github.com/bitcoin/bips/blob/master/bip-0032.mediawikiArmory does not yet support BIP0032, but the current implementation is similiar.
