I'd suggest having a conversation with your boss where you ask for advice about how he wants you to delegate tasks and who he wants you to delegate tasks to. Something along the lines of
"Hey, boss. I know when we talked at my review, you suggested that I learn to delegate. I'd love some advice about how to do that without stepping on any toes.
I've got task X (relatively small, relatively self-contained) that I'd love to get off my plate. I think (name of delegate) would do a great job on it. But I know that (name of delegate) is senior to me so I don't want to come across as asking him to do my work for me. Can you help me figure out how best to approach (name of delegate)?"
This shows that you listened to the suggestion. It shows that you've identified what you want to delegate and who you want to delegate it to, which is a good first step. And it engages your manager to assist you in navigating the political environment (which will likely be different in different organizations and teams).
This may lead your manager to sending out an email to the team letting them know that you're going to be delegating work to them. Your manager might tell you that he'd rather you delegate the task to someone else or putting some additional parameters on the suggestion (e.g. "Jane has been asking to get involved in that project, why don't you work with her instead" or "I've already warned Tim and Steve that they're going to need to pick up some of the tasks you been doing, give them whatever you'd like"). Plus, people generally like it when others come to them for advice (particularly when they can show they've put in some effort already).
Depending on the team, the organization, your relationships with the team, it's also possible that you can just walk up to a senior team member and ask them to take on a task. If it works, that's probably an even better option. But that's also something that is very hard for folks on the internet to diagnose-- the reaction will be very different if your team members see you as a really competent contributor that is being groomed to be a team lead, whether they see you as just "the new guy," whether they see you "jumping the line" to be the team lead or whether they see you as the natural choice, etc.