I know some has been mentioned about Rogue Traders having a Writ of Claim on planets or systems and that the Imperium doesn't care about how they govern these systems just that the tithes come in as scheduled. What does this mean for the Rogue Trader?

To be more specific, the Rogue Trader in my game has a Lost Dynasty and has discovered that his ancestor founded a system in the Calixis Sector during the Age of Plunder. If the ancestor had a Writ of Claim, does this mean he can just come in and take what resources he needs? Or is he still needing to buy these resources and can mine the natural resources from the uninhabited worlds uncontested?

The Writ of Claim would only apply to those planets outside of Imperial Space - once an area is termed Imperial space, the Imperium will usually place it's own governors and the like on them.

Unless you or a member of your family is still the governor of the planet (if the Imperium decided to leave your ancestor in charge, or another member of his family), no, you'd have to requisition/buy them as normal.