Levy abruptly stepped down as chairman on Thursday after urgings from the club's owners in the morning.
Asked for his immediate reaction, Poyet told Oddspedia: "I was trying to think from the moment it came out. I don't think we will know. I would be surprised if we know why, the truth. Daniel has improved the club in terms of infrastructure, the stadium and the training ground, the change of quality was tremendous, amazing. At the same time, during many, many years, the team was not able to win a trophy from 2008 until now, which frustrated many people.
"I always say that it's very difficult to spend the money in all the other things and no need on the team or to do in a perfect way. To be able to win titles and transform the club is always difficult. But I would say two things are for me key. One, winning the Europa League, 100%. Everybody was asking him to win something. Okay, I won something. And a European tournament, which is tough.
"Secondly, 25 years. I don't know if it has something to do with the number as well, but sometime we put in our life, you know, 25 years or 50 years. Like I said, I don't think it's negative or tremendous either way. The club will stay. When they sack a manager, they say the club and the supporters, they continue.
"He's a chairman going, so I don't think we should give him more than he deserves. It was great for the club in many, many aspects. He was criticised a lot and he took it. I think he knew very well what he was doing. Every time he was tremendously under pressure.
"He made decisions, took over his responsibilities in changing managers, in a very clever way. He was never involved in any public issues. I think he's managing himself very well and if not, he wouldn't be there for 25 years."
Careful what you wish for
On the fans who have been pushing for Levy to be ousted, Poyet has warned about being careful what you wish for.
He also stated, "I would say to the fans, I hope what is coming in is not worse. Because then maybe you're going to have to start singing Daniel Levy’s name to get him back. The good thing for the newcomers and the new chairman and the new people in charge is that all that part that I said to you was magnificent training ground and a stadium is done.
"So, now, if there is a change, I suppose there is going to be more focus on the team, which, like I said, we'll see. Many, many people spend money, and only one or two win things every year. So, let's see if they can do it."