This is one of several psychological dramas involving Robin and the death penalty. Dick Grayson learns that one of his schoolmates is angry at Batman because his father is about to be executed based on work by the Caped Crusader. But the boy knows his dad couldn't have committed murder:
But when Dick confronts Batman about this, he gets rebuffed oddly enough to raise his suspicions: Dick goes down to the Batcave and reviews the case, where he learns that Brent (the father of his classmate), Denton (the murder victim) and another man named Walsh were partners in an air freight business. Brent and Denton often quarreled about finances and one day Denton was found dead in the office, with Brent standing over him in a daze with a golf club (the murder weapon) in his hand.
Robin naturally suspects the remaining member of the partnership, Walsh. But it appears that he did not gain financially by the death of one partner and the pending execution of another: However, it's obvious from the company Walsh is keeping that he's got some sort of racket going on. Robin pokes around and discovers that he's smuggling wanted gangsters out of the area. He's captured by some of the crooks and brought in to face Walsh again, who admits that's why he killed one partner and framed the other. He wanted to make sure that they didn't catch on to his other activities. Meanwhile, Robin is shocked to learn the time: Robin gets free and basically disables the crooks, including Walsh, although Batman and the cops also arrive just in time to apply the finishing touches. Batman had realized all along that Brent didn't really commit the murder, and so the whole conviction and execution was staged. Brent's son, Jimmy, and Dick weren't told because they had to act naturally or Walsh would realize it was all a plot to make him reveal the motive behind the murder/frame-up.
The ending is pretty typical of these tales: Incidentally, that rather gripping scene on the cover (which is reproduced on the splash page as well) never occurs in the book; it's apparently a bit of artistic license. My guess is that it's an homage/swipe of this famous Airboy cover from about a year and a half earlier: