Hi,
The url you gave wasn't showing the picture, but sniffing around on Google, I found the pic you were talking about
The sketch in question was from Sesame Street's first season, and photos from the sketch were featured in The Sesame Street Book of People and Things, which was published in 1970. Underneath the photos were typed lines of Ernie's dialogue.
The sketch was essentially Ernie telling Bert that he was waiting for an important call, and explaining how "the Monster" came in and asked to use the phone quickly. "The Monster" kept talking and talking and talking, and Ernie was getting more and more anxious because he was waiting for the call, he tried bribing "the Monster" with a cookie, but after eating the cookie, he then proceeded to eat the phone. When the phone was in "the Monster"'s belly, that's when Ernie's imporant call came.
Like many other Muppets, Cookie Monster appeared on the show as a generic character who was not yet fully-developed, and at the time this sketch aired, he was not yet "Cookie Monster", but all the essential CM elements were there. That is why, in this sketch (and in another one featured in the same book), he is referred to as "Monster" or "the Monster", and not yet by the name "Cookie Monster".
This is not the only time that we've seen Cookie Monster eating a telephone. In Christmas Eve on Sesame Street, there is a running theme in which CM is trying to write a letter to Santa, asking for cookies for Christmas. The idea of receiving the cookies he is asking for excites him so much that he eats the pencil he was using to write the letter. He later tries to write the letter with a typewriter, which suffers the same fate as the pencil. In a last-ditch effort to make contact with the North Pole, he tries to call Santa to request cookies over the phone, but guess where the receiver ends up before he gets through.
Cherry "My name would be a great cookie flavour" Pizza