One thing to remember here is that RSA can be used in 2 ways: to encrypt and to sign.

If RSA is used to encrypt (for example if you send an encrypted message using PGP) then factoring directly breaks the encryption.

In practice, a lot of encryption on the Internet uses RSA to sign the hash of a key obtained using Diffie-Hellman. In this case breaking RSA would allow the NSA to impersonate but not directly break existing communications. The problem with impersonation is that it is very noticeable.

What I find odd about the linked article is that it only talks about factoring. In practice, the discrete log. problem is just as important and is very much related to factoring.