An Area Gate is a barrier in Octo Valley and Octo Canyon to block the player from advancing to the next area before all missions in the area are complete.
Area Gates have wires with lightbulbs attached. The outlet of the wire is attached to the boss mission and gets discharged when the mission is complete. As a result, the lightbulbs explode and the passage is cleared.
Trivia
In Area 4, there is an exploit to skip the area and proceed to Area 5 by inking the passage and jumping from the side of the Area Gate.