To "fall on your sword" means that you are accepting responsibility for some negative act (typically of a negligent or criminal nature). Historically in some societies military commanders would commit suicide by "falling on their sword" to maintain some honor after losing a battle. Just as a comparison, "throwing someone under the bus" is a term you may have heard that is the opposite of accepting personal responsibility. This is where you try to divert attention away from your own wrongdoing by talking about the possible wrongdoing of other people, or how other people were the main cause of a problem.

It means to accept defeat; or to go to extremes to indicate one's defeat. As Dugup said, it originates from the ancient practice of soldiers committing suicide this way rather than being captured. Another eg' So, because I lost the contract, am I supposed to fall on my sword or something?'