The DBConnection pooling mechanism in Elixir's DBConnection library allows for efficient management of database connections. When a query is sent, the first available connection is found by looking at the ETS queue for a holder reference, and ownership of the holder ETS table is passed to the calling process. The connection is then used to make the query, and once complete, the holder reference is returned to the ConnectionPool process, which updates the ETS table and transfers ownership back to itself. If all connections are in use, queries are added to the ETS queue until a connection becomes available. The pool size determines how many connections are spawned under the DBConnection.ConnectionPool.Pool supervisor. Increasing the pool size can improve performance but also increases resource consumption.