Walk along a constraint until a level curve just barely touches it. That kiss is the answer.
Slide all the way around and watch the two arrows. They line up only at the marked points, and those are the only places worth checking.
Drag the point around the constraint curve.
You want the largest value of f, but you are not allowed to wander freely — you must stay on the constraint curve. So walk the curve and watch the level curve of f that passes through you.
Almost everywhere, that level curve crosses the constraint. And a crossing is always bad news: it means one way along the curve takes you to higher ground and the other way takes you lower, so you cannot possibly be at a maximum or a minimum. You can still improve.
The only places you can't improve are where the level curve stops crossing and merely touches — where the two curves are tangent. Tangent curves have parallel normals, and the normals are the two gradients. That is the whole method: look for the points where ∇f = λ∇g. The multiplier λ is just the ratio of their lengths, and it is negative when the arrows point opposite ways.
The graph underneath is the same story told as a one-variable problem: f restricted to the curve. Its peaks and valleys sit exactly at the marked candidates, which is the point — the constraint has turned a two-variable problem into a one-variable one.