In Physics SO the intuition of the Laplace operator (divergence of the gradient) is explained by resorting to the finite difference version: the Laplace equation is satisfied as long as the value at a vertex is the average of the surrounding values, akin to the explanation in Wikipedia of harmonic functions.
Or the more enjoyable blog explanation, motivating them through minimal surfaces: soap bubbles,
or the 3Blue1Brown video: Much like a minimum with a positive second derivative, a positive Laplacian in a 2D surface would indicate local minimum (concave up) in the way that the neighboring points on average are higher in value.
Professor Strang gives a rather artistic impromptu intuition right here proposing a grid with unweighted edges, and noticing that for internal vertices, the degree would be $4,$ corresponding to second differences with the surrounding unit-value edges, but this is a bit shaky in what it is really happening in the example. Clearly the average of adjacent entries in a Laplacian matrix doesn't really work because of the embedded degree matrix in the diagonal:
Others approach the graph Laplacian by noticing the connection to Newton's law of cooling.
Is there a better "visual" to see the intuition of the graph Laplacian?