I heard about this sentence,
"How to do it well? The fine point: ...."
What does 'the fine point' mean in this context?
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Ordinarily a fine point means a subtle distinction (sense 8a in TheFreeDictionary).
The fine point isn't a standard term in any context, but per @Robusto's comment, there is the fine (or small) print meaning (usually unfavourable) details buried inconspicuously in small print at the end of a verbose contract. In common parlance it's often assumed that such apparently minor details may in fact be of crucial importance when deciding whether to sign a contract.
In fact OP has mistranscribed "The fine points" (at 1m 45s) in an instructional video. As @Spare Oom indicates, a fine point is a subtle or abstruse point. Used in the singular that would often imply an obscure, peripheral detail, but in this particular context the speaker is using the plural, and his meaning is...
The subtle details which must be mastered in order to swim really well.
This doesn't seem like a particularly common usage to me, possibly because of the unwanted negative associations with fine print, and a fine point in an argument, which is often said dismissively of some valid but peripheral assertion. In OP's context there are no such connotations or other ambiguities, but I wouldn't advise using the expression more generally unless your context is equally clear.
The details about how to do something well can be referred to as the fine points or the finer points. See Thesaurus.com under the first entry of detail.
Also, in the context of an argument, the fine point is something subtle or abstruse.
There is also an idiom, (not) to put a fine point on something, to which the example may be referring, though it is less likely and without the rest of the context it is difficult to tell. The example from The Free Dictionary says:
something that you say when you are going to say exactly what you mean, even if other people may not like it. Well, not to put too fine a point on it, it's entirely your fault.
The trouble in this phrase is the definite article. To say "'the' fine point is" without a contextual scale effectively insinuates there is no finer point, which is absurd unless one is referring to a geometric point, which is, by definition, infinitely small.
However, compared to my previous, verbose discourse the finer point of my argument is that context matters.
To say "not to put too fine a point on it" insinuates that the nuance being declared may seem excessively precise, but the sharpness of the distinction is relevant and worthy of attention.
To say "the finer points" insinuates that there are more and less nuanced degrees of substance; some more general, some more specific. A general point of swimming may be to "propel through other". In contrast to this general nuance, finer points of swimming may deal with swimming through a mass of water, or people, or thoughts, etc. An even finer point in the context of water might be related to the degree and finesse in technique using one's body, whether for endurance, speed, synchronization, grace, power, depth, etc., or to techniques in using technology, e.g., fins, weights, compressed air, etc.
Not to put too fine a point on it, but context matters, as so many others have already stated.