Expressions used in Everyday Spoken English in Social and Professional contexts
When you 'wrap up' something, it can mean to cover something in order to conceal or protect it. You might 'wrap up' in warm clothes in the winter to protect yourself from the cold. If you buy someone a present in a shop the cashier might ask you if you want the item 'wrapped up', in a box or wrapping paper. You can wrap up food that you intend to eat later on or to protect it from flies or the temperature. Similar in some contexts to 'gift wrap' or 'to cover'.
To 'wrap up' can also mean 'to finish or conclude something. Similar in meaning to 'wind up'. You can 'wrap up a meeting' or 'wrap up the details' of something or even 'wrap up a deal'. If you become 'wrapped up' in something, it means you are absorbed by it and fail to care or think about anything else.