This pattern means 'not have to do something' or 'can get away without doing something.' It often carries a sense of relief or convenience—something that could have been necessary is avoided. The verb 済む originally means 'to end' or 'to be settled,' so the nuance is that the situation is resolved even without doing X.

It is commonly used in spoken and written Japanese, but the full form なくて済む is a bit neutral; in casual conversation you might hear なくて済んだ (past tense) when talking about something that didn't end up being needed. The pattern is also often used with 済む in past or conditional forms, like なくて済んだ or なくて済むように.

Be careful: the negative form must be the ない-stem plus くて, not the て-form of the negative (which is なくて, but that is exactly what you use). However, you cannot use ないで済む interchangeably? Actually ないで済む also exists with a similar meaning, but なくて済む is more common in standard speech. For nouns, use でなくて済む, but note that this can sound slightly formal; in casual speech, じゃなくて済む is more natural.