You can do four things with Missionaries: Defame, Worship, Dispell, or Missionize.
Defaming will lower the spirituality in the enemy temple. Click a temple/city and select "Slander" to defame. This is influenced by the demoralization attribute.
Worshipping with Missionaries will give increased benefit to the temple/Holy City. Examples include faster rate of spirituality gain (Prayer stat), bigger range (Conversion Stat), and losing spirituality at a slower rate when defamed (Piety stat). You must assign missionaries to the temple/Holy city for the effect to take place. See the Temple section [TRTT] for more info.
Dispelling is basically attempting to send an enemy missionary back to its city. You click the enemy missionary, click "Dispell", and select a missionary. If the missionary succeeds, the enemy missionary will leave. If it fails, nothing changes.
Missionizing will simply give your missionary experience, as well as add the that certain square's spirituality.