The intuition behind the boundary operator, as well as the intuition behind d2=0 are tightly linked.
Imagine a triangle, edges labeled AB, AC and BC. What do we expect its boundary to be? The edges, in some order. If we label the vertices A,B,C and make the edge labels correspond to the vertices in the obvious manner, then we can give each edge an inherent orientation by pointing it towards the letter that occurs later in the alphabet.
And thus, going once around the triangle, we list our edges as AB + BC + CA = AB + BC - AC, using the sign to indicate when we're backing up along a directed edge.
Extending this notion of a boundary, as a bona fide geometric boundary, we notice that it ends up being linear - if we have several simplices that glue together, the resulting boundary is going to be an appropriate signed sum of the boundaries of each simplex, with the connecting edges cancelling out by the sign choice. And we can figure out a formula - the one with alternating signs and dropped vertices - that generalizes the 2d idea of a boundary to something that both looks right and works.
Now that we know the boundary of a triangle? What's the boundary of the sequence AB, BC, CA of edges?
It shouldn't take all that much to come up with the idea of "It doesn't have one". Which is, essentially, the idea behind d2=0: the boundary of a boundary should vanish, because that's what it does if we think of the geometric situation and try to formulate an intuition.
From this point on, though, a lot of algebraic topology and more importantly a lot of homological algebra takes the intuitions we build by considering easy examples, and runs with it; in homological algebra without even a resemblance of regard from the geometric origins.