Basically you collide a hash like SHA-1 or MD5 by getting it into a state where transitions don't twiddle as many bits, and then smashing the remaining bits by brute force trial. But, such states are weird so from inside the hash algorithm you can notice "Huh, this is that weird state I care about" and flag that at a cost of making the algorithm a little slower. The tweaked SHA1 code is publicly available.

If you're thinking "Oh! I should rip out our safe SHA256 code and use this unsafe but then retro-actively safer SHA1" No. Don't do that. SHA-256 is safer and faster. This is an emergency patch for people for whom apparently 20 years notice wasn't enough warning.

In theory the known way to do this isn't the only way, but, we have re-assuring evidence for MD5 that independent forces (probably the NSA) who have every reason to choose a different way to attack the hash to avoid detection do trigger the same weird states even though they're spending the eye-watering sum of money to break hashes themselves not just copy-pasting a result from a published paper.