The Explorer trophy consists of 5 sub-challenges:
Abandoned Mine: Explore the Abandoned Mine.
From the Town Map, select Abandoned Silver Mine and choose Explore Mine.
Beach Cave: Explore the Beach Cave.
From the Town Map, select Coastal Cave and choose Explore Coastal Cave.
Lighthouse: Explore the Lighthouse.
From the Town Map, select Rock's Edge Lighthouse and choose Explore Lighthouse.
Lost Knowledge: Find and read a book lost in the depths of Sugar Maple Coast.
In the course of exploring the four locations involved in this challenge group, your Sims will find various objects including books. This challenge is a little tricky because not every book you find will complete this challenge. People have reported completing the challenge after reading different books but the book that worked for me was "I, Timaeus" so if you find this book in your inventory, try reading it.
Mausoleum: Explore the Mausoleum at night.
At night, click on the Gilman Hill Graveyard in the Town Map and choose to Explore Catacombs.