Simple intuition
Some states are perfectly aligned with a measurement. If the system is already in that special state, the measurement gives a definite answer.
Precise explanation
If |a⟩ is an eigenstate of operator  with eigenvalue a, then measuring the observable associated with  returns a with certainty. In circuit language, gates are unitary operators, while measurements are associated with observable operators.
Example or analogy
Analogy: if you shine a flashlight straight at a wall, the beam already points in the wall-normal direction. An eigenstate is a state already aligned with the quantity you are checking.
Common misconception
Eigenstates are not rare mathematical curiosities. They are the states that make measurement outcomes definite, so they sit at the center of the theory.