How do offshore bank accounts function?

Same as any other bank account, with the exception being, on how the reporting might be done with certain countries, i.e. disclosing account information.

Offshore accounts are essentially structured on the basis that you are essentially opening a bank account in another country, thus, in many instances, escaping the taxation that you would otherwise be subjected to in your home country.

Because your country laws do not apply outside your borders, off-shore accounts thus are able to by-pass various personal and/or corporate laws regarding, money, disclosure, taxation, pension, permissible investment, holding, distribution, investments, etc. that you would otherwise be subjected to in your home country.

Do note, it may be illegal for you to open an offshore account, depending on your country of residence or your principal country where your business is registered.

All in all, offshore accounts are used to take advantage of a different country's banking services and low (or zero) taxation rate and escape the taxes in your own country.