What are these red stems and the things that look like roots? Do they damage the plant and how should I take care of it?
They are roots.
The plant is perfectly OK. In the wild they propagate by bits breaking off, growing roots, and starting a new plant. From the picture it looks like you have pruned some stems off, so the plant probably thinks more stems are going to get broken off soon and has started growing some roots in anticipation of that.
Many succulents are naturally coloured red, but you can't see that because the chlorophyll they produce for photosynthesis is green and hides the red colour. If they are in bright light, they don't need so much chlorophyll and parts of the plant may "turn red".
It's nothing to worry about. If you don't like the red colour, move the plant to where it gets less light.