What are these faint light brown stains on the white stucco ceiling.?

Unless the shingles are 3D and are guaranteed to last 25, 30, 40 or 50 years then it's time for a new roof. What you are seeing are water marks from the rain. Get a few different roofers for an estimate, as this problem will continue to worsen. Sorry about the news. Look at the shingles, are they curling at the edges, missing, you can tell by looking at them if they need to be replaced.