Users tend to use an application in unexpected ways. Conventional happy path testing is not sufficient. Event with failure path testing only a subset of the possible ways to work your way through an application are tested.
Exploratory testing is a disciplined method to test (corner) cases in the application.
Listen to your curiosity Define what you're curious about and explore ASAP.
Despite the "exploratory" nature of the tests, it is required to do some preparation upfront. The expedition should be prepared, you have to define a goal of the exploration. No need to wander of the path: take notes about alternative paths that can be investigated in future expeditions.
Parts of the system, or flows through the system, that are not commonly checked are tested. Possible issues can be resolved and tested with automated tests.
Create a simple Test Charter Card which describes your curiosity with the following format:
Explore: A target With: Resources To Discover: Information