From the Wikipedia page - current version at [1].

The text "The Magic Words are Squeamish Ossifrage" was the solution to a challenge ciphertext posed by the inventors of the RSA cipher in 1977. The problem appeared in Martin Gardner's Mathematical Games column in Scientific American. It was solved in 1993–1994 by a large joint computer project co-ordinated by Derek Atkins, Michael Graff, Arjen Lenstra and Paul Leyland. More than 600 volunteers contributed CPU time from about 1,600 machines (two of which were fax machines) over six months. The coordination was done via the Internet and was one of the first such projects.

Ossifrage is an older name for the lammergeier, a scavenging vulture that is famous for dropping animal bones and live tortoises onto rocks to crack them open. It might perhaps be considered among the least squeamish of creatures. The 1993-1994 effort began the tradition of using the words "squeamish ossifrage" in cryptanalytic challenges.

The difficulty of breaking the RSA cipher — recovering a plaintext message given a ciphertext and the public key — is connected to the difficulty of factoring large numbers. While it is not known if the two problems are mathematically equivalent, factoring is currently the only method of directly breaking RSA. The decryption of the 1977 ciphertext involved the factoring of a 129-digit number, RSA-129, in order to recover the plaintext.

Ron Rivest estimated in 1977 that factoring a 125-digit number would require 40 quadrillion years, even with the highly conservative assumption that modular multiplication could be carried out in a nanosecond; he therefore then believed that RSA-129 could never be broken in practice. What he failed to take into account was the possibility of progress in factoring algorithms, and a lot of progress was made in the following decades. Atkins et al. used the quadratic sieve algorithm invented by Carl Pomerance in 1981. While the asymptotically faster number field sieve had just been invented, it was not clear at the time that it would be better than the quadratic sieve for 129-digit numbers. The memory requirements of the newer algorithm were also a concern.

There was a US$100 prize associated with the challenge, which the winners donated to the Free Software Foundation.