Ziyech will move to London at the end of the season for a fee rising to €44m.
The 26-year-old, who has agreed a five-year contract with the Blues, was speaking to Ajax TV.
"Happy, proud, I am excited and I cannot wait," he said.
"It [Chelsea] is a big club in a big competition. I like the style of play, they play really attacking football, and that is something that fits me and that is one of the biggest reasons I chose therefore."
Ziyech was asked what part Blues manager Lampard played in him opting for Stamford Bridge for the next part of his career.
"A big role. We had a lot of contact in the last couple of weeks and he gave me a good feeling and that was also big to do with the decision."
Ziyech also explained in the video interview what his aims are for the remaining few months he has as a player at Ajax.
"First of all to enjoy it with the people here [at Ajax], the team-mates and the people we work with, the trainers and everybody around the club. The most important is to enjoy it and win the trophies we have to win."
Watch the full interview below: