What does "clutching at straws" mean?

It means “looking for a way to do something which is actually hopeless and unattainable”. Like trying to stop yourself from falling by grabbing onto something that’s not going to hold your weight.

Often, it’s used to describe taking an untenable position in a discussion – trying to justify something by coming up with propositions that just don’t make sense.

It means “grasping at straws”, which is the more common variation on the idiom. From the linked source:

to depend on something that is useless
to make a futile attempt at something
trying to find some way to succeed when nothing you choose is likely to work
trying to find reasons to feel hopeful about a bad situation

The idiom originated with Thomas More’s Dialogue of Comfort Against Tribulation (1534).

Imagine that you are about to fall. Imagine that you grab / grasp / clutch at a bunch of straws to avoid falling. It is a desperate move, because the straws will never be able to hold your weight and you know it, but you do it regardless (because it is your last hope).

The expression above could be translated as:

"It is almost impossible for me to imagine a situation in which a woman will not be offended by being called a..."

It indicates desperation. A drowning man will clutch/grab at anything, even at straws (this is the (older?) usage that means 'dry grass' not drinking straws) in an attempt to save himself. I have no idea what @tony gil is talking about incidentally. :-)