It implies that the person escaping was being pursued or watched, and they managed to avoid capture or detection.
Here are some examples:
* "The thief gave the police the slip by blending in with the crowd."
* "After the party, I gave my date the slip and went home early."
* "The dog gave its leash the slip and ran off down the street."
Essentially, it means to "get away from" someone in a cunning way.