Its meaning is, literally, 'the things that should have been changed having been changed'. (They are thus a very good example of why Latin is said to be a concise language.) More helpfully, they might be rendered as 'having due regard to the different circumstances in different contexts'.
Don't use the phrase mutatis mutandis at all unless you are happy that you do understand it. When you are happy with it, use it mostly to explain why you are using an idea, usually a comparison, which you have developed in one area in a separate area.
When you have described the game of Association Football, you might say 'The same is true, mutatis mutandis, of the different forms of football played around the world'. If you are an engineer, you might be talking about structures ('The same is true, mutatis mutandis, of the framework of all buildings', after you have discussed timber-framing). If you are a linguist, you might be discussing grammatical features: you might have described the noun phrase in English, and then say 'The same might be asserted, mutatis mutandis, of other European languages'. In these examples, the engineer is saying that all frameworks behave in the same way even when made of different materials - once one has made allowances for the specific nature of each material. The linguist is saying that broadly speaking the noun phrase is the same in European languages - once you ignore their petty details.