The setup process wasn't very intuitive and didn't work reliably; a lot of trial-and-error was involved. But once it's set up, you can just leave it there and not worry about it. Just log into zendesk.com/chat/agent to chat with your customers.
On a separate note, we want to caution anyone considering subscribing to Zendesk's services. Their business practice isn't great; we informed them of our intention to cancel our subscription before the renewal date, but they didn't process our request and our credit card was charged. When they finally processed our request a few days later, they issued a full refund; however, as we're in Canada and they charged us in USD, there was a change in the foreign exchange rate in the days between when we were charged and when we were refunded, and in the end we lost over CAD $700, and Zendesk would not take any responsibility.