A high street is the main road in a traditional British town, where most of the big-name shops would be found (and, historically, grand houses, civic buildings etc). A high street bank branch would be one you could walk into like a shop, where you could make deposits, open an account, talk to an employee and so on. In recent years there's been a sense that high streets are in decline and shops are being closed: partly due to the internet taking over and partly due to overall poverty, corporate penny-pinching and national decline. It's very common to hear people say of banks that it's near-impossible to speak to a human, because high street branches are closed and everything is supposed to be done online.
I've never heard of a drive-through bank at all, but I wasn't alive in the 60s and 70s.
