According to the official documentation on installing the DSpace 6 REST API, what you need to do is this:
Installing the REST API
The REST API deploys as a standard webapp for your servlet container / tomcat. For example, depending on how you deploy webapps, one way would be to alter tomcat-home/conf/server.xml and add:
<Context path="/rest" docBase="/dspace/webapps/rest" />
In DSpace 4, the initial/official Jersey-based REST API was added to DSpace. > The DSpace 4 REST API provides READ-ONLY access to DSpace Objects.
In DSpace 5, the REST API adds authentication, allows Creation, Update, and Delete to objects, can access restricted materials if authorized, and it requires SSL.
Disabling SSL
For localhost development purposes, SSL can add additional getting-started difficulty, so security can be disabled. To disable DSpace REST's requirement to require security/ssl, alter [dspace]/webapps/rest/WEB-INF/web.xml
or [dspace-source]/dspace-rest/src/main/webapp/WEB-INF/web.xml
and comment out the block, and restart your servlet container. Production usages of the REST API should use SSL, as authentication credentials should not go over the internet unencrypted.
Which of these steps have you done? Did you restart your servlet container (eg tomcat)? Then what do you use to test the REST API, and what happens -- you need to show the actual URL you use and the response.