I was invited to the online welcome event for my cohort. There supposed to be some greetings from faculty and current students as well as Q&A session. Apart from that what should I expect from this kind of events? Would I be required to say some words about myself or how I am happy to join the program or smth? I'm not that great of a speaker regarding social gatherings and a bit nervous that will be asked to say something I am not prepared for.
In the pre-pandemic world, such meetings, face to face, had the purpose of encouraging people to accept offers, to "personalize" the faculty, to let students and prospective students meet one another informally. They gave new students the opportunity to meed possible advisors and discuss, if only for a few moments, mutual interests, both research and personal.
And yes, they were a bit uncomfortable for incoming students who didn't really have a great idea of the place or their future there.
In the pandemic era, the zoom meets are an attempt to capture as much of that as possible under unfortunate circumstances. It isn't intended as an inquisition.
Yes, you might be asked about your interests and maybe a bit about your background, both personal and professional. But think of it as the sort of thing that might happen if you bumped in to someone who didn't know you well.
It is impossible to say exactly how it might go, since these meet ups are new to everyone involved now. If you are uncomfortable, I'd suggest that you just try not to be the first to talk, so that you get an idea of the kinds of things that go on. But, I don't think any preparation is needed or expected. In the US, at least, it is enough to be able to say that you are interested in some general area (Analysis) or in some fairly broad sub-field. You don't need to come, certainly, with a dissertation topic in mind. Human qualities may actually be more important than deep research qualities at such a meet. There is plenty of time to do the hard work once you start.
There won't be a "sorting hat" as in the Harry Potter books.
And take the good advice that astronat gives in a comment to your question. Find a way to participate.