It is used when something good may happen, but if it does happen, there will be some bad things that may result with it. "The new road comes at the cost of a major portion of the old woods that had to be cleared for it." "The new austerity package comes at the cost of sever short-term job loss, due to the resulting slowdown." "The battle was won, but it came at the cost of numerous casualties on both sides."

When a decision is made, at the risk of something worse in the future. So something good may happen which seems beneficial to begin with, but in reality this is detrimental to something else! "Buying a holiday home may come at the cost of leaving the family" you can also say "at the expense of"