Basically, it means the marriage is over because one (usually) or both (sometimes) of the spouses want it to be over.
As used in divorce actions, the terms irreconcilable differences and irretrievable breakdown are big boxes into which much can be placed. What is salient about their application to divorce actions is that their use eliminates the need for assigning blame for the failure of a marriage. Generally the test of irreconcilable differences is that the marriage, by agreement of both parties, is hopelessly broken and without hope of reconciliation.