Basically:
People who fight together in-game get cutscenes showing them getting closer as friends. This increases their support rank. Characters who have a higher "support rank" help each other out in fights more. This means your characters will get free attacks if they're standing near each other (or on the same tile) and automatically block each other's attacks (taking no damage). Higher support rank not only makes this more common, but also increases the stat boost you get by having people paired up during a fight.
If they're opposite genders, they can get "S" rank support. This gives you a dual attack/defend rate of about 80%. S rank is treated the same as being married (you get an engagement cutscene), so each person can only have a single S rank they're attached to. You can get A rank support (which is around 50% rate) with basically everybody else that they're able to be friends with, but only one S rank with an opposite gender.
You can manipulate this by pairing up particular people together in fights. If you force two people to fight together non-stop for a few chapters you can get them to S rank pretty quickly. Remember, if you S rank, they become married automatically and you don't get much of a say in it. You can prevent that by never selecting to watch the S rank cutscene.
Characters who get S rank and get married will have children. Later in the game you can recruit these children to your group. Their stats and available classes are always a combination of their parents, and max out higher. So depending who you pair together, either naturally or on purpose, can determine what kind of stats your later characters have.
This entire thing is optional though, except for one thing you might want to be aware of: (Spoilered, but only minor.)
For story reasons, Chrom will automatically marry whoever he has the highest support rank with (that isn't dead or already married off to somebody else) at the end of Chapter 11. If there's someone in particular you want to pair him up with, make sure you plan it before then.
So yeah, Kio just likes playing dating sim:
