The transaction brigde is what MS-DTC (Distributed Transaction Coordinator) uses to manage transactions across several systems, is an implementation of the Two Phase Commit. Basically, its used internally by Windows to manage a common transaction across several systems, something like this:
Take a C# application running on IIS.
You enter a TransactionScope, and delete some records from an SQL Server instance. Now (still in the TransactionScope) you connect to Oracle, to insert some rows.
At this point, you have an open transaction in each RDBMS, but both transactions are combined into the Distributed Transaction by DTC.
When you commit your TransactionScope, a commit will be issued to both systems, and only if both systems have to successfully execute the commit for the distributed transaction to finish correctly. On the other hand if you issue a rollback, both systems will roll back their transactions.
The trasaction brigde is what its used to communicate with those other systes that understand the 2PC (Two Phase Commit) Protocol.
Hope this helps.