Sheaves capture two properties: if you have a solution to a problem on a big piece of a space, you can shrink it to a smaller piece, and if you have solutions on small pieces of a space that agree with each other on overlaps, you can glue them together to get a solution on a bigger piece.

An easy example is a function on a set. If you have function defined on the whole set, you can shrink it to give you a function defined on a subset. If you have functions defined on several subsets, and those functions agree on the overlaps of the subsets, then you can use that to define a function on the union of the subsets. More interesting examples arise in topology and related fields.