6 ft 1 in, often written as 6' 1", is the written form of 'six feet and one inch'.
That's not what we say in conversation, however. In conversation, or as part of a longer text, I'd read that simply as "six foot one".
Why 'foot' instead of 'feet'? I've no idea, but that's what feels natural for many people, myself included.
There's also no need to say 'inch' or 'inches' (in the case of plural) : when we're talking about height, it goes without saying that the second number refers to inches.
How tall is Tom Cruise? He's five seven.
In other words, 5' 7".