Exploration lets you send an expedition into one of Chillverse's maps to earn XP and, sometimes, a rare Artifact. It isn't something you control move-by-move in real time — instead, you pick a specific location within a map (called a chamber), spend energy to start the expedition, and then wait for it to finish. Once it's done, you come back and collect your reward.
Each map is split into 5 chambers that unlock in order — you'll need to finish one chamber before starting the next one within that map, so it's worth planning which chamber you tackle next rather than jumping around.