This phrase is used when someone is trying any possible means of getting out of a difficult situation. It usually means a person is saying or doing things with only the slightest chance of success.

Be aware that while the explanations about the origins are correct, you will find that the phrase is now often used in a sense of “you have lost the debate and are now just bringing up desperate arguments”

It comes from an expression about a drowning man. He needs a life preserver or a branch of a tree or even a small piece of wood but his situation is so bad that he is grasping at straws, something that we know is of little use.