I wouldn't try to explain the mathematics of the backdoor.  Just explain that the NSA hid a secret backdoor in there.

Instead, I would suggest focusing on the history and the context.  For instance, you could explain about Crypto.AG, how they spiked their RNG to help the NSA spy on their customers.  You could explain how random number generators are a classic weak point in cryptography, because of the RNG has been backdoored, whoever knows the backdoor can recover all your keys.  You could also explain how this is very hard to detect: since RNGs are supposed to output random-looking bits, if it's actually pseudorandom bits that the NSA can predict but no one else can (if the bits are actually coming from a PRNG that the NSA knows the seed to, but no one else does), this is very difficult to detect through black-box testing or through inspection of the supposedly  random bits.

If you wanted, you could explain that back in 2006, cryptographers discovered the mathematical structure that would allow a backdoor but shied away from making any accusations, because, hey, surely the NSA wouldn't do that -- that would be so egregious a breach, that it seemed hard to imagine it actually contained a backdoor.  And now things are changed: it appears that what seemed unimaginable, is in fact reality.  So, you could explain how this is an eye-opener for the cryptographic community and how it will change the way we think about government standards and system design for the future.

You could then explain the impact.  You could explain how the impact is believed to be relatively minor, because most people heeded Bruce Schneier's warning from 2007 and didn't use Dual_EC_DRBG.  However, you could explain that at least one major crypto library (RSA's BSAFE) does use it, for reasons that are unclear, so this might have an unknown impact on an unknown number of deployed products.  You could explain how this backdoor is likely to only allow the NSA to decrypt traffic, not anyone else.  And finally, you could explain what the biggest impact is: the potential for loss of trust in cryptographic standards and in American companies.  I'm not sure that we've seen the general public lose trust yet -- I think people still trust Google and Apple and Amazon to have their back -- but if that ever happened, that might be hard to recover from.