Things that are touchy are hard to handle — not literally, but to talk about. Touchy subjects make people upset.
There are a lot of touchy topics in life. Religion is a big one, since people have such different yet strong beliefs. Racism and sexism are touchy. Politics can be touchy. Asking someone how old they are or what they weigh are very touchy questions. Something is touchy if you need to handle it with kid gloves — or maybe avoid the topic altogether. People often use euphemisms — soft, evasive terms — for touchy subjects, to make discussing them easier.