The practical attack enabled here is mostly around digital signatures. An attacker could produce documents A and B that both have the same SHA-1. They can then get someone to sign document A (which really is signing the SHA-1 of document A), then use the signature with document B and make it look like they have a document B signed with a valid signature.
As an example, if document A is a regular SSL certificate request, and document B is a "CA certificate", the attacker can trick a real CA into signing a rogue CA into existence, which can then sign its own certificates that will be trusted by every browser. This has already happened with MD5 in 2008 [3].
[1] https://en.wikipedia.org/wiki/Collision_attack