K1 was encrypted using a particular form of a Vigenere cipher known as Quagmire III. This is a fancy way of saying it was a Vig cipher with two keywords.
The first keyword was KRYPTOS. This part was not so hard to guess. It was the name of the statue, and the ancient greek word for hidden. Furthermore, the right hand side of the statue is more or less a large Vigenere tableau. This was a huge hint to use Vigenere in the first place. The tableau contained the keyword KRYPTOS repeated over and over many times.
The first keyword in a Quagmire III gives you the order of the alphabet for the encryption. In this case, the keyword KRYPTOS gives us the alphabet: KRYPTOSABCDEFGHIJLMNQUVWXZ. This part was not very hard, was it?
The second keyword was PALIMPSEST. A palimpsest is a manuscript written on parchment or vellum that has been scraped off and then used again. Again a word very fitting to the theme of the Kryptos statue.
Each letter in the second keyword in Quagmire III gives you the starting letter for each alphabet used in the encryption. This part is difficult to describe with words. The Realm of Twelve website (see the links topic) gives a great visual explanation for all of this.
If you know the method of encryption, solving for the keys with frequency analysis is not that difficult. In fact, there are methods for determining if an encryption is a Vigenere in the first place. But the clues where there on the statue in plain sight.
Solving this section took 9 years.