Shitty old wallets would screw up their transaction signing, and reuse a value that is supposed to be unique per signature. The reuse of this value can be detected and used to extract a private key.
Discussion
This is the same principle used to crack PS3 discs back in the day. Sony made a fucky wucky and reused the same R value across all licensed titles when they signed the digital assets for the purposes of DRM (the PS3 would only play discs signed by Sony's private key). Once diligent hackers had detected the she R values across multiple titles, they were able to extract Sony's private signing key.