The basic idea is that for a provider (Github, Google, etc), to keep their Safe Harbour protections, they have to act like a 'dumb pipe' and just obey these notices. Just like how the phone companies trying to keep their Common Carrier status[1].
Even without the DMCA, if the service provider interjects itself into the dispute they risk legal action against themselves. For example, they could evaluate the claims wrong, and get sued to allowing actually infringing material to stay up.
It goes like this:
1. DMCA is filed by a person/entity with the service-provider.
2. Service-provider takes down offending content.
3. The person/entity that submitted the offending content now has the option to file a counter-notice with their service provider claiming that they feel/know there is no infringement. If they do file a counter-notice, then we continue on. If they don't file a counter-notice, then the story ends here.
4. The service-provider notifies the person/entity that filed the DMCA notice that a counter-notice was filed. The service-provider can now restore the content, but there is some weird stipulation that they need to wait something like 10 ~ 14 days before doing so.
5. The person/entity that filed the original notice now has to take legal action against the person/entity that they feel has infringed their copyrights if they want anything more to happen.
The party that files the DMCA notice stipulates that they believe that their work has been infringed under penalty of perjury.
[1] For a long time, people on the Internet believed that 'Common Carrier' provisions applied to ISPs, but it really only applies to phone companies. The Safe Harbour provisions of the DMCA are meant to give something similar for ISPs, though there are different rules.