This is an expression that is mostly used in American English. When you say `let the chips fall where they may', what you mean is, do whatever you have to do without worrying too much about the consequences of your action. In other words, do the right thing and don't worry about the fallout. Let things happen naturally, do not make an attempt to try to control everything.
Example:
The minister said that he would speak from the heart, and let the chips fall where they may.

The `chips has nothing to do with the chips we normally eat. This idiom comes from the world of logging. When you chop down a tree using an axe, every time you hit the tree, pieces of wood (chips) scatter. While you are cutting, you do not worry about the various chips flying around; you don't really care where they land. As a cutter, you remain focused on the task at hand which is to chop down the tree.