Stealthily describes HOW someone does something: quietly and secretly, moving in a way that is meant not to be noticed. If you walk stealthily, you are watching your own feet, keeping your steps soft, perhaps holding your breath — because being seen or heard would spoil everything. A cat crossing a room towards a bird moves stealthily. A child creeping downstairs at midnight to look at the presents moves stealthily. So the word says two things at once, and both matter: quiet, and on purpose. Quiet by accident is not stealthy. There has to be a reason for hiding. It is a somewhat literary word. You meet it in novels, in news writing, and in storytelling much more than in speech; in conversation people say 'quietly', or 'without anyone noticing', or 'he sneaked out'. That does not make it a difficult word to use — it is a lovely word — but if you say it in a casual chat, it will sound slightly grand, which may be exactly the effect you want. It is often mildly negative, or at least suspicious, because people who hide their movements usually have a reason. But not always: a nurse can move stealthily through a dark ward so as not to wake the patients, and there is nothing wrong in that. Writers also use it about things that are not people. Prices rise stealthily. An illness spreads stealthily. Here it means the change is slow and quiet enough that nobody notices until it is large.