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From this video, just listen to the first 8 bars (first 15 seconds). It sounds like the chord progress is I III7 II7 V# V7 in F major scale to me.
I don't know how to explain the function of this V# major chord. It sounds so natural but I don't know why.
The closest thing I can think of is the backdoor progression which is VIb VIIb I. But in this case, there is no VIIb after VIb. The fifth chord is not even close to it.
Assuming your transcription is correct, that is not a ♯V chord, but a ♭VI chord. This is borrowed from the tonic minor and provides a powerful pull to the dominant (because of the half-step-downward resolution of the ♭VI chord's root, third, and fifth).
With Roman numeral analysis you can put sharps and flats on the Roman numerals to show the root is altered.
But, while a altered dominant root with ♭V makes sense a ♯V does not. Here is why...
Usually such alteration are about borrowed chords or mode mixture. For example, in a major key you can have a ♭VII borrowed from mixolydian. When you add flats in the order of the circle of fifths the sequence of altered tones, starting from C major, is B♭ E♭ A♭ D♭ G♭ and the corresponding modes in sequence are mixolydian, dorian, aeolian, phrygian, locrian. When you add sharps the sequence is short, it's just one move F♯ taking it to lydian.
When you get to the end of either the flats or sharps direction the next step would alter the tonic from C to either C♭ or C♯. Things get weird, but if we keep going up with sharps to finally get to the alteration of a ♯V, which starting from C major would be a G♯, we first move through C# locrian, then C# phrygian.
So, alterations like ♭II or ♭VII, for example, make sense, because you can say they are borrowed from the parallel modes phrygian or mixolydian. From C major those would be C phrygian or C mixolydian. But for ♯V you wouldn't even get the root of the chord until C# phrygian. Where the tonic has changed! But, even then the chord would be #vo, a diminished triad. To get a diatonic #V you have to add seven sharps to get to C# major!
All this becomes a whole lot clearer when you use the key signature which should be given with Roman numeral analysis. This... C: I III7 II7 C#:V C:V7 ...uses a G# major triad but requires a really clumsy tonic change in the middle of the progression. But, if you spell the root as something diatonic to a C tonic, it will be enharmonic equivalent A♭ or ♭VI. This... C: I III7 II7 ♭VI V7 ...is so much clearer! One tonic with borrowed chords.
Basically, you shouldn't normally see root alterations on I or V. They don't make sense. A ♭V would be a diatonic chord in locrian, but locrian is pretty unusual. Not at all common.
Alteration on dominants that you will commonly see are not alterations to the root, but on the chord's fifth and ninth, ex. G7♭9 or G7♭5. I've seen that combined with Roman numerals like V7♭9.
(Expanding on Dekkadeci's answer) the bVI7, sometimes with a few modifications, is a common predecessor to the V chord (perhaps through the I64 to avoid parallels or just to extend the piece). These are the "Augmented Sixth" chords (with geographic names).
There are two musical "tricks" happening, the half-step move from the b6 to the 5 stands out; using the bVI7 also resolves by a half step. (Usually, the flat seventh is written as an augmented sixth, Ab-C-Eb-G#, to show the direction of motion.) The usual modifications (which do not change the function, just alter the color and voice-leading) are to drop the fifth (Eb in this case) or to lower that fifth which gives two tritones. This is all covered in (Common Practice Period) harmony books.
