In order to complete this quest, the Vault Dweller needs to visit the Vault 13 library, at the far end of the south-west part of the level. Once there, they need to use his Science skill on the computers in the room.
With a successful Intelligence check, he will gain 350 XP for researching important information. Failing the check will just produce a message about surfing the Interweb.
The skill can be used on both computers in the library, as the experience stacks.