In isolation, FAC doesn't belong to any key. It's just an F major . When it's put into a piece it goes into context. Usually, a piece IN F will end ON F, and feel that it has ended, without the need for any more to follow. Often, that piece will start ON an F (usually in the first full bar if there's an anacrucis) as well. The feeling of 'home'.
At various other points in the tune, cadences will appear. They will give clues as to whether the song is IN F. If they are Fs themselves, and the song is indeed in F, those points will feel that the song could finish there. If they are, for example, imperfect cadences, they will give the feeling that there is more to come. For example, a middle 8 ending on CEG makes it feel the need for the next part to continue back on FAC. This is known as V-I, the 'I' being F in this case.