Vassals are an essential way to control your empire in Crusader Kings 3 without micromanaging every little detail. Some of the regulated titles do not require your direct attention. Sometimes it is better to give it to a loyal official. Once you've broken up your kingdom into enough sections that allow you to remain the primary ruler, how do you expand your empire and claim more territory?
Before going into the details of expanding your empire, it's important to understand how to appoint vassals to guard your territory. Go to the kingdom tab, the green crown at the top right of the screen, open it to view your kingdom, then tap the domain page. It contains a list of all the domains your character owns under their name. If your domain limit is too high, gift these slots to your vassals. You want to make sure that you don't give too much to a vassal, so that they don't become an independent character and remain under your rule. Check again which areas are under the same De Jure to create a new title, which is given to your ruler, and you can pass it on to your heir once your main character dies.
Now, to add more vassals to your empire, you have the choice of using an aggressive expansion strategy or using a more user-friendly approach. Both are viable options. The aggressive approach requires you to do everything possible to invite yourself in and use your advisor to fabricate claims on your behalf. It may take a while for them to use this method, and they might only be able to expand into one territory, one at a time. To increase this method, you can use the decision to invite claimants to your court. It costs 350 prestige to increase the chances of prestigious guests entering your yard. Still, it's more likely that someone will claim a powerful De Jure nearby, which could give your kingdom more territory.
The alternative is to befriend the vassals nearby. Most of the time, your territory and kingdom will be much larger than your neighbors. You can take this opportunity to convince them, thereby increasing their overall opinion. For those who have a leader with the Diplomacy lifestyle can become friends with another character, thus improving their relationship. If you have a strong enough army and are good enough friends with a nearby vassal, they can offer themselves to you, or you can see in the predictions what would happen if you offered to make them your vassal. When they accept it, they join your empire, expanding your region.
Both of these methods are viable and it pays to keep your options open. It all depends on your rule. For example, a ruler with the Intrigue lifestyle might find enough hooks on a potential vassal to force him into his kingdom. There are several options, and without a land shortage in Crusader Kings 3, there are always vassals who can join your cause.
