About Creating Web APIs in java

The exact and official answers should be found on Jersey site's front page "About".

I'll try to explain it from a different perspective and a little bit more specific.

Jersey calls itself a framework, but I think it's easy to understand that it's an extension that makes servlet more convenient than a framework.
Even if you use Jersey to develop a web API, you will still be creating a web API with Tomcat+Servlet.

For example, if you consider providing identity-based search for user information such as /user/[id],

public class User{

    private long id;
    private String name;

    // getter, setter, ...
}

If you implement it with the original servlet, it will look like this:

@WebServlet("/servlet/user/*")
public class UserServlet extensions HttpServlet {

    @ Override
    protected void doGet(HttpServletRequest req, HttpServletResponse response)rowsServletException, IOException {

        long id = Long.parseLong(req.getPathInfo().split("/")[1]);

        User search using User user=//id

        resp.setCharacterEncoding("UTF-8");
        resp.setContentType("application/json");
        resp.getWriter().append(toJsonStr(user)) .close();
    }

    private String to JsonStr (User user) {
        return new StringBuilder()
                .append("{")
                .append("\"id\":").append(user.getId())
                .append(", ")
                .append("\"name\":\"")
                .append(user.getName()) .append("\"")
                .append("}").toString();
    }
}

If you incorporate a feature extension called Jersey, it looks like this:

@Path("/user/{id}")
public class UserResource {

    @ GET
    @Produce(MediaType.APPLICATION_JSON)
    publicUseruser(@PathParam("id")longid){

        User search using User user=//id

        return user;
    }
}

This is an example, but

If you look it up online, you can see many samples using a library called Jersey

The reason for this is that many sample makers thought it would be better/easy to use Jersey to write like the latter, although it could be done using the original servlet like the former.

Should I use some kind of framework?

Generally speaking, whether or not a framework should be used is determined by the balance between the advantages of the functions provided by the framework (for me) and the disadvantages such as learning costs and costs.
I think the sample makers are introducing Jersey because they think that it is OSS and the learning cost is relatively low.

Supplemental:
Jersey is not the framework in the nofollow noreferrer">narrow meaning (Jersey does not provide control inversion).
If you're too obsessed with the word framework at this stage, it might be a hindrance to your understanding.