To send your companion on an exploration, click on its portrait to the left of your avatar's profile photo:
Under the image of your companion, you will find the places accessible to you for exploration:
In each location, the blue points represent the available explorations. Each exploration costs between 10 and 40 energy points. The length of exploration the possible rewards vary.
When you click on a blue exploration point, a window will appear that explains the place of exploration, the energy cost, and the length of exploration. It is then possible to send your companion on the exploration, or you can explore immediately by paying in Gold Coins.
At the end of the exploration, a window will appear informing you what your companion found during its exploration:
It is possible that your companion won't find anything, the odds are influenced by its Luck Bonus!