They sure do. Anyone in your court applies their stats to your nation and can have big impacts on resource generation. A court scholar will give a very nice boost to research, a soldier to Training production, etc. If you hover your mouse over those resources at the top (gold, training research, etc.) you can see where they are derived from. Note that characters can have negative stats that negatively impact your resources! A character with a ton of Courage might be great for Training but if he has negative Discipline then he’ll lose you some gold every turn, that sort of thing.
In addition, these are your available characters to assign roles to. They can lead armies as Generals or fill positions in your council like Chancellor, Ambassador, Spy Master, etc. They are also the people that you can make Governors of cities to apply various bonuses there. For things like court positions and governorships, they require a tech to unlock (and for the latter the construction of a Garrison improvement in the city radius).
Note that not all characters are eligible to fill all roles depending on their archetype (Judge, Builder, Hero, etc.). Someone might be able to lead an army into battle but not be able to be Chancellor. When you go to assign these roles, it will show only the characters that can fill those slots. For your children you can get events that might grant them certain archetypes and the UI should show you what positions it would make them eligible for.
EDIT: “How do I do it?”
To assign a general, select your unit/army and there’s an action to assign general there. For Governors, you need to research the tech that allows you to construct a Garrison (it’s an improvement workers build). Once it’s built the Governor can be assigned via the City screen (look towards the bottom left). Pro tip while you’re in that area of that screen: You can also send luxuries to that city to help keep its happiness up!
For council positions, those are in the upper right of the screen. You can see portraits of people filling those positions (at the beginning this might just be your Consort and your Heir). If you’ve unlocked the tech for Ambassadors and do not have it filled, the icon for the Ambassador position will be glowing/flashing up there. Click on it, then select the character.
Once a character is in that position, you can then click on their portrait and have them do tasks for you. To stick with the Ambassador example, you can click on his/her portrait and then send them on a Trade Mission to another Civ. Or you can send them off to try to craft Peace treaties between your nations, form alliances, etc.
A note on terminology that might be confusing at first: Your default diplomatic state is Truce. This is different from Peace. Think of a peace treaty kind of like a non aggression pact in other games in the sense that it’s a stronger form of peace than the default “we’re not at war” Truce state.