1. When a minor marries someone of age, their spouse becomes their legal guardian until they become of age (or become emancipated, which is a difficult process generally). Why is that an issue? Well, in abuse situations, it can be incredibly hard for the minor to leave their spouse who is their legal guardian. It's not much different to runaway children being returned to their parents, even when their parents are the source of the abuse. Certain states are worse for this than others, so called "reunification states" where they will absolutely try and return you to your legal guardian unless there's clear and convincing evidence why they shouldn't;
2. A lot of child marriages happen so an adult can escape a statutory rape or rape charge as prosecutors don't tend to prosecute when it's a spouse. It should be noted that spousal rape only became a crime in all 50 states of the US around 2005 (IIRC) and this only really picked up steam around 1990 or so;
3. Because of this lack of power, it creates a whole host of other problems, such as exploitation to the point of trafficking.
Politically, preserving child marriage is a uniquely Republican issue [1].
[1]: https://www.rollingstone.com/politics/politics-features/repu...