Why does this happen?

The person who is being comforted is trying to hold it together, which simply means trying to get their emotions under control. The gesture of a hug tells them they are not in it alone, there is someone else there to help them hold it together and so they cry, it is a release of pent-up emotion.

Well for me personally, on general level, that doesn't happen, when someone comes into the situation I usually stone up a little, and I generally find it hard to express emotion in front of people.

However, when it's a common sadness, as when you and that 2nd person are sharing the same sadness, I think it's partly because you're feeling the sadness for yourself, but you're also feeling the sadness for the other persons experience, Like when someone close to both of you passes away, you're feeling sad because that person is gone, then you're seeing that other persons sadness and also feeling sad for them to be experiencing the loss as well.

Just what I think. :)

because the person that should be caring for her does not

Because they make you feel more sorry for yourself

God knows best.