so as not to break or damage壊さない means “not break” or “not damage,” and ように expresses purpose or precaution: “so as not to...” In context, it can also mean “so you don't wear yourself out” when referring to one's health.
JA: このグラスを壊さないように、ゆっくり運んでね。
EN: Carry this glass slowly so you don't break it.