That is not how you get/use the key per se and the video covers it quite well* but anyhow
* I am assuming you have seen it but for the benefit of others reading
http://www.youtube.com/watch?v=5E0DkoQjCmI - 35 minutes 30 seconds starts the crypto explanation, 37 minutes and a few seconds starts the where Sony screwed up section. Do watch the whole thing though.
Encryption and data signing is usually based around a so called hard problem (actual mathematical term like real, imaginary, complex, logarithmic and the like) and the two big ways to do it (as least as far as asymmetric stuff goes in the real world) is to use prime numbers (RSA is a good one here) or the maths behind ellipses/elliptical curves (ECDSA being the choice method here).
http://security.ece.orst.edu/koc/ece575/papers/ecdsa.pdf gets right into it (mainly section/chapter 4 although you might want to read 3 as well).
One of the equations used in the signature is along the lines of
signature part which remember is just a number = ( sha1 of thing you want to sign + (private key multiplied by result of hard maths function) ) / random number
The hard maths function also uses the supposedly random number and indeed is the only true unknown as far as the stuff the person has to decode it knows. However being a hard function you can not take the result and the other known input data and use it to get the random number.
The random number used in the hard part is the same as the equation mentioned up above and the private key is the private key which should be constant for your entire implementation or else there is no point.
All well and good until you consider that Sony chose to use the same random number every time they did that signature (there are multiple different signatures/private keys for various parts and versions of the PS3 platform but they are all broken in the same way so it is just a matter of repeating the process)
Getting two signatures is easy enough (there are multiple programs after all), you have the data being signed and sha1 is a known function if nothing else.
s1 = signature 1 and s2 = signature 2 k= random number m1=program1 m2=program2 r= hard maths part d= private key aka holy grail of PS3 hacking.
s1 = (m1's sha1 + d*r)/k
s2 = (m2's sha1 + d*r)/k
As Sony screwed up and used k twice (or indeed all the time) you can combine the equations to lose the private key*hard maths part (remember r is derived using k so the same k value makes for the same r value). If they had used a different random number it would have been impossible/pointless trying to combine them.
s1-s2= (m1's sha1 - m2's sha1)/k
Rearranging to find k is easy enough.
While you are at it rearrange the "easy" equation to have the private key "on one side".
private key = ( (random*signature) - sha1 )/ result of hard maths
As part of the whole signature you have the result of hard maths (it is part of the public key), you have the random number thanks to the trick above and the signature and sha1 values are there as well. The private key is now a matter of perhaps 30 seconds on a calculator.