As mentioned in the episode, Mark Vernon recorded a series of lectures on Aristotle's philosophy of friendship (listen to the lectures on iTunes). These were published in 2006 in conjunction with his book, The Meaning of Friendship. As stated in the Stanford Encyclopedia of Philosophy's account of Aristotle's view, the source material in Aristotle seems to be Books VII and IX of the Nichomachean Ethics.
Here's Vernon being interviewed about the book:
Watch on YouTube.
Here he is on Philosophy Bites talking about it, and here's an article from the Guardian about it. Quoting from that article:
The word idiot derives from the Greek idiotes, "a person who lives only for themselves". For example, in his Funeral Oration, Pericles praises individual initiative in ancient Athenians, but only inasmuch as it contributes to the greater good: an idiot has no right to be part of the city-state, the collective upon which their happiness depends...
...Aristotle's conception of the friend as being "another self". Post-Enlightenment, we take this to mean that a close friend is a mirror of your own self, someone with whom you find personal resonances, thereby realising that though autonomous you are not alone; there is someone quite like you and who you quite like.
But for Aristotle something more connected, more dependent is being described. The sense of self of the two friends is, in fact, one. Like the two eyes, their operation is conjoined. I do not find out who I am in solipsistic, narcissistic isolation, but in my friendships. I can no more be a full person without a friend than a magnet can exist as a single pole.
I find that focusing on this issue of friendship provides an easy way into the larger issues of the Politcs; this account of friendship is the other side of the coin to the fundamental master-slave relationship that we mentioned as providing such a motivation to Hegel and Nietzsche. Another good point of comparison is Rousseau, whose model of society building out of the extension of family relations has much in common with Aristotle's account.