Talking to the current leaders of each faction, you can ask for their side of the story. If you belonged to the faction that the claimant is trying to overthrow, the fiefs you currently own convert to the rebel faction of your claimant (temporarily under your control). If you were starting your own kingdom, you have to give up your kingdom to become a vassal of the claimant. With over 200 renown, the claimant will allow you to swear an oath of vassalage and your allegiance changes to the rebel faction.
If you have less than 200 renown, they will tell you that you need to gain more. Once you find a claimant, they will tell you their side of the story and will ask you to take up their cause.
