Smoothly and with no trouble at all. It is used almost only about how a plan, a meeting, a project, or an event turned out: 'The interview went swimmingly.' The image behind it is a fish moving through water — no effort, nothing blocking it. Notice how narrow this word is in real life. It nearly always sits with the verb 'go': things go swimmingly. You will rarely hear anyone say a person swims or works swimmingly. It also has a slightly old-fashioned, cheerful flavour — the kind of word a happy older British speaker might use — so younger people often reach for it playfully, or with a touch of irony. That irony is worth knowing: because the word sounds so bright and confident, people love to use it when things went badly. 'Well, that went swimmingly,' said after a disaster, means the opposite, and the tone of voice tells you which one is meant. Nothing here is about actual swimming.