Why do the British use the word "flipping" for emphasis?

In the English (British) TV drama, Coronation Street, the word "flipping" is often used to stress a situation, so much so that it feels like a swear word to me to some degree:

• I've got a flipping headache


• That flipping moron

I think usually people from US like to use the word "freaking" or "fricking" instead.

I know "freak" is a very strong word, but can't understand why "flipping" can be too. Is it because the term "flipping someone off" (a term I just learned while searching for answer myself)? And I am still a bit confused why "flipping someone off" has the meaning of, you know, "flipping someone off".

Answer

Words such as fudging, freaking, fricking, and flipping are euphemisms for fucking. Here's an entry on "flip" (my emphasis):

flip (v.)

1590s "to fillip, to toss with the thumb," imitative, or perhaps a thinned form of flap, or else a contraction of fillip (q.v.), which also is held to be imitative. Meaning "toss as though with the thumb" is from 1610s. Meaning "to flip a coin" (to decide something) is by 1879. Sense of "get excited" is first recorded 1950; flip (one's) lid "lose one's head, go wild" is from 1949, American English; variant flip (one's) wig attested by 1952, but the image turns up earlier in popular record reviews ["Talking Boogie. Not quite as wig-flipping as reverse side--but a wig-flipper" Billboard, Sept. 17, 1949]. Related: Flipped. Flipping (adj.) as euphemism for fucking is British slang first recorded 1911 in D.H. Lawrence. Flip side (of a gramophone record) is by 1949.
(Etymonline)

It's not exclusive to BrE, as it's also heard in AmE. However, it might be less common.

They are used just like you said (for emphasis) and they're used when offensive language is not allowed or not called for. They seem like swear words because they are, just milder ones.

It has nothing to do with "flipping someone off."

As the the entry above suggests, "flipping someone off" likely comes in part from the fact that flip/flipping can be used to describe the movement of extending a finger in that manner.