I have not used Zelle, but have used Venmo, which is similar.
You set up a bank account to which and from which you can transfer money. There's an app for that.
If someone sends you money, the app alerts you, and then you can transfer to your own bank from your Venmo "account." There is no fee to do so, and it's usually in there at the beginning of the next business day. There are different settings, so you can ask the transaction to be public (appear on a common "feed") or private (only visible to you and the other person involved).
The only "problem" I've ever had is that someone I didn't know sent me money! It was for rent, and she'd simply pulled up the wrong name, as the intended recipient's and my names are very similar. I was at work, and she messaged me (there's a place to do that) to ask if she could have her money back. By the time I was able to see my phone, she'd panicked and contacted an administrator, from whom I'd received a friendly polite note to ask my "point of view." I sent her back the money and then replied to the administrator that I'd done so, delayed by my inability to check my phone at work, and received a nice thank you.
Other than that, I've never had an issue. My kids and I use this to transfer money frequently.