I thought we already invented a technology that can work without the internet. I was shocked for a second after reading all the replies. It seems it's just usual thing, you synch up when you are online. Thats all.
If you are using your wallet on an air-gapped device, you don't need to connect it to the internet to see your update balance, all you need to do is to create a watch-only version of your wallet, and in it you can see everything, create transactions and also broadcast it, your air-gapped device is to remain disconnected from the internet forever, and would only be used for the purpose of signing transactions when you want to spend.
But if you are going to send Bitcoin yourself, you really need to have internet, you can't have it because the transaction won't work without an internet data connection for sure.
You can do it without an internet connection if you are using an air-gapped wallet and a watch-only version of the wallet:
https://electrum.readthedocs.io/en/latest/coldstorage.html