Expressions used in Everyday Spoken English in Social and Professional contexts
‘To find a way out’ of something is to literally escape from a situation or to exit somewhere. Figuratively, it means to find your way out of a responsibility or an agreement. To ‘talk your way out’ is to use your skills of persuasion to get out of a difficult or undesirable situation. Similar in meaning to ‘having the gift of the gab’ or ‘weasel out of something’.
Something considered ‘way out’ is unusual or very strange or progressive. This is a slang phrase that comes from the US and is similar in meaning to ‘far out’. It is very informal and used in social contexts.