What’s the point of invited talks?

As someone who has given a fair number of talks and has now had the "pleasure" of giving an online talk, I can say that the advantages of in-person talks include the following.

• You can have informal chats with people from the audience. One of the main problems of online chatrooms is that there usually can be only one discussion at the same time. This precludes informal chats like this because no one wants to monopolize the only communication channel. Also there is a whole paraphernalia around talks, you go to lunch with interested people or whatever and can have more talks. If you're really treating them as job talks, then you want to talk about stuff other than research with people. Departments don't want to hire people who cannot integrate well; being someone that is pleasant to be around is a big plus, and you can't prove that if you just come, give your talk, and leave.
• (In the same vein, giving a talk in person is the opportunity to visit the university/department and see if you'd enjoy working there.)
• If it's a conference (i.e. >1 people come to talk) then you can meet the other presenters. It may be a nice way to meet people who have the same research interests as you. Also sometimes a good way to arrange meetings with existing collaborators who work far away.
• You can meet people who don't come to your talk. Sometimes people are busy and aren't free at the precise time of your talk, but you can run across them in some corridor or something and talk with them. And somehow the busiest people are the ones you often want to meet, for obvious reasons. Maybe the department's big shot has ten meetings that day, but you can still catch her for ten minutes and leave a good impression.
• Usually around the talk I would have long conversations with some people (usually the one who invites me) about our respective current research. Sometimes this can even lead to collaborations. It's hard to do online because this sort of discussion is "open ended" in some sense: you just talk about whatever goes through your head at that moment. I don't think anyone would consciously schedule an online conversation to do this, but it's extremely valuable.
• People are apparently afraid to ask questions during talks. During a normal talk I usually have lots of questions during and after; at my online talk and the ones I've witnessed, much fewer questions are asked.
• Giving an online talk is just awkward. Not many people enable their webcam, and nobody other than the presenter has their mic on. It's very difficult to know if your audience is reacting positively or negatively to what you're saying. It's very destabilizing.
• There are also technical things. Having pre-prepared slides is good, but there is always some moment where you want to write something more on the board that you hadn't thought of, or some picture, or some motion you want to show off using your hands. It's difficult with an online talk.
• Lastly, and some people may not want to openly admit this because it's a bit selfish, but if you enjoy traveling and visiting new cities, being invited to do it is always nice.

Of course a lot of this is a bit on you. I was once an insecure PhD student, and I tended to just stay hidden in my hotel room the night before, in the guest office if they had one the day of the talk, give my talk, not dare talk to anyone, and leave. Don't do that. Take advantage of the fact that you're there in person.

I would say that the only advantage of online talks is the ease of planning. As someone who also organizes a weekly research seminar, inviting someone is now as simple as shooting them an email. The other person just has to think about whether they want to give a talk that day or not. No travel to think about, no hotel no expense reimbursements, no administrative hurdles, no nothing.

I would assume that if someone has invited you to talk, they have found your research interesting. The talk is, perhaps, intended to be just the first contact in what might develop over time into a rich collaborative relationship.

I think that would be especially valuable to a junior faculty member. If you were a well established, senior, member of the academy they are more likely to be saying that they want to be associated with you and your reputation. For a junior member it is the other way about. We think you can add something here.

A circle of collaborators is one of the most valuable assets of a faculty member, especially one on the way up in the academy.

There is another possibility, however. Some places will have a young faculty member who doesn't yet have such a circle, and the existing members of the department aren't in fields closely enough related to provide support. They might invite people in just to support that person and give them both ideas and potential colleagues. I've seen this happen in practice, actually.

Others have made excellent points already, but I might want to add an additional insight from the student perspective.

Q: Why should one travel around to give talks at universities?

To add back a little bit of diversity and to highlight other approaches.

Some universities / departments have a very strong bias towards one end of the spectrum (in terms of who they hire, what they are teaching). Invited talks are a great opportunity for students to learn something about what the rest of the world thinks / researches. Economics is a good example for this, as there are at least two bigger approaches to the whole subject. I know of a department where students are allowed to take a course that solely consists of invited talks (those talks are basically picked by students of the later semesters) to introduce other approaches and topics. This course is in high demand, and is very well received by the students.

With this answer, I want to highlight the giving aspect of the topic.

I think the gaining part is nicely highlighted already.