A 'run-in' is an argument, or some other trouble, with someone else. It usually implies you had a chance encounter with them as well.
I had a run-in with my landlord. He really wants his rent.
'Picayune' is a somewhat uncommon word that means 'of little value' or 'petty.' In this context, it is being used to emphasize the argument between Schumer and McCaskill (the run-in) was of little consequence and should not have happened to begin with.
The encounter (run-in) is what spoke volumes. 'Speak volumes' is an idiom that basically means to reveal a great deal of information.
The way a person dresses speaks volumes about their character. <--- the clothes you wear reveal information about your personality.
So, the article is saying that this petty argument, and the fact that it happened at all, gives insight into the overall anger and frustration felt by the Democrats.