Like Shmoop, Brain is a super-computer with the personality of a child. And that is how Brain solves a problem that no other computer could solve.
Unlike most of our other important robots, Brain is more like a computer than a robot—he's just a brain (41). But since he has (or, ahem, is) a positronic brain, he has a personality, like the other robots we've met in these books (30-1). So, whereas US Robots has emotional robots, their rival Consolidated only has thinking machines (30). Unlike those thinking machines, which broke down when trying to solve the hyperatomic drive problems, Brain can solve the problem of how to travel through hyperspace precisely because he has a personality. And by the time we've reached this story, the idea of robots with emotions and personality should be no surprise. So when Calvin says something surprising, we're not surprised when Brain goes "Gosh" and "By golly" (47, 49). Well, we might be surprised because that's so kid-like; but we're not surprised that Brain is human-like.
Now, Brain is useful to this story because he can do the impossible—solve hyperspace—by doing something else impossible—killing people. Of course, he can only "kill" people because it's temporary; because Susan Calvin told him that we don't mind being dead (48); and because he finds a sense of humor as an escape valve (322). This makes "Escape!" kind of a strange story: Calvin figures out what's going on, but even if she didn't, nothing bad would happen to anyone. Which is just a nice way to remind us that robots are on our side.
In fact, "Escape!" really shows us how necessary robots are to us; that is, Robbie might have helped take care of Gloria, but really, any nursemaid would've helped, even a human. But Brain does something that only he could do.
What does this robot tell us about robots? He tells us that robots are necessary to our technological progress. Brain also reminds us that emotions and personality are an important part of science and progress. And he reminds us how annoying some little kids can be.