Context
This is a question mainly about Lucene (or possibly Solr) internals. The main topic is faceted search, in which search can happen along
Faceting
There are two answers for faceting, because there are two types of faceting. I'm not certain that either of these are faster than an RDBMS.
Field Cache. This is just a normal (non-inverted) index. The SQL-style query that is run here is like:
select facet, count(*) from field_cache where docId in query_results group by facet
Again, I don't think this is anything that a normal RDBMS couldn't do. The index is a skip list, with the docId as the key.
Multi-term search
This is where Lucene shines. Why Lucene's approach is so good is too long to post here, but I can recommend this post on Lucene Performance, or the papers linked therein.