Robertson has joined Spurs on a free transfer after nine trophy-laden years at Liverpool, linking up with his new teammates ahead of the club's pre-season tour of Australia and New Zealand.
"I'm excited to be here," he said.
"It was a bit different in terms of the signing being announced and having to go to the World Cup, but my first day has been good, getting to meet people, forgetting people's names (laughs) and trying to work my way around the training ground and things like that.
"It's perfect for this time in my career and that's why when the opportunity came up to join, I was excited to go for it and now that I'm here, I'm excited to keep progressing in my career and keep doing good things. Hopefully I can do that in a Tottenham shirt. This is a massive club with an amazing training ground, amazing stadium and a big fanbase – I've played here plenty of times and I know that the supporters get right behind their team.
"It's a club that's going places. The last couple of years from a league point of view have maybe not been up to the standards that the Tottenham fans expect, last season they showed a lot of tenacity to get to the end and stay up, but now, even being in from day one, there's a huge determination to not allow that to happen again. It's up to us to make sure we're better and to make sure that doesn't happen."
De Zerbi, appointed manager in March, is one of the main reasons Robertson chose Spurs.
"The manager is a huge pull," he said. "I've had chats with him, I've had a phone call with him and both him and his staff seem so good at what they do and how they want to play.
"It was a great phone call. We know of each other – he knows me as a player, I know him as a manager and it was just more that we were excited to work with each other. He's well known in world football for being a fantastic manager and the fact that he's here, hopefully I can do really good things under him."
Robertson believes his experience can benefit the club's younger players.
"Obviously now I'm a lot more experienced, I've got a lot of games under my belt and hopefully I can use that experience to help the younger lads and the other lads as well," he said. "I think everyone connected to this club just wants Tottenham to have a successful season and I'm no different to that. Hopefully I can play a big part in that and play the way I know I can.
"I can already see from day one that everyone is determined to bring the best level of their game and hopefully then that can produce special things for us this season."