Mongo Replication with VIP
Community Wiki answer generated from comments on the question by James Wahlin and Markus W Mahlberg

James: The arbiter is a voting member in an election but doesn't appoint members as primary or control the process. I would suggest reading: Replica Set Elections in the MongoDB manual.

When there is an election, all client connections are dropped by MongoDB. At that point in time the client will connect to one of its seeds and perform an isMaster call. This call reports back to the client which member is primary and which members are secondaries. In that way the client will seamlessly write to the correct member, based on read preference.

Markus: It will route requests from the application to the secondary. The drivers are replica set aware. Since all connections are either actively dropped by the old primary or stale, the driver will try to find the current primary in the list of replica set members it knows. You can verify this behaviour by setting up a replica set locally, running at 3 different ports on your local machine and kill one of them at random. After the short while an election needs to be finished, your client application writes to the newly elected primary.