What does raising a characters affinity level do?

It gives that character bonus stats (any version of it) so if you have lvl 15 affinity on megumin for example, ANY megumin you use will benefit from the stat bonuses. these bonuses might be magic atk and magic resist for certain characters or physical attack and defense for others. Also at certain affinity levels, their ultimate ability levels up, improving the damage or status effect of those abilities