This progression is a modification of the common IV iv I cadence.
As @tombeatz says, the ivᵒ♯7 it is a modification of the iv chord. Flatting the fifth adds the leading tone of the scale, causing it to sound more dominant than iv. The ♯7 is a pedal tone that occurs throughout the composition.
♯ivᶱ7 is a IV with a raised root and a dominant seventh. The dominant seventh is just that pedal tone again, but the raised root I like to think of as a suspension, even though it is not the third. V7/IV makes us expect a IV, which is almost what we get - just with a suspended root, and even though the following chord still isn’t a IV, the root does resolve when it drops the semitone.
EDIT: I didn’t understand what @dnuttle meant at first, but I get it now and it is a good insight: If we ignore the ♭9 from ivᵒ♯7, what remains is just III, aka V/vi. Even though the next chord is I7 instead of vi, it still works because they are relatives and share most of their notes.
It is a creative progression, combining the IV iv I cadence with this suspended root thing, as it results in the perception of constantly falling notes but with lots of forward momentum.
More on the suspended root thing:
It is pretty common to raise the root and even other notes of a chord as sort of a suspended passing chord that then resolves. If there is a term for this, please let me know! I think of them as sort of extended suspensions. Here are some other examples:
It's a Jungle Out There by Randy Newman Chords, Melody, and Music Theory Analysis - Hooktheory :
V7/V iiᶱ43 V7 and V7 vᶱ43 V7/IV
Alone Again (Naturally) by Gilbert O'Sullivan Chords, Melody, and Music Theory Analysis - Hooktheory :
v7 iiiᶱ43 V7/ii
Dream A Little Dream Of Me by Doris Day Chords, Melody, and Music Theory Analysis - Hooktheory :
V7/ii iiiᶱ43 V7/ii
Even raising every note in a chord sort of sounds like a suspension; try I V7/V ♭VI7 V I.
As for the notation of the two borrowed chords:
The one just labeled (bor) is actually a lydian ivᶱ7 and could be labeled (lyd). Some half diminished chords (specifically ♭iiᶱ7 and ♭viᶱ7) are unreachable in Hookpad, but this isn’t one of them.
The chord from phrygian dominant is kind of an anomaly in that it may be one of the only two diminished chords with a ♯7 reachable in Hookpad. Only the harmonic minor and phrygian dominant scales have a minor major seventh that can be modified with a ♭5 to make a ᵒ♯7, so the fact this chord is even possible to enter into Hookpad is kind of a fluke. If I’m wrong and someone thinks of another way to write more of their transpositions, I’d love to see it.
EDIT: I originally wrote there was only one ᵒ♯7 chord available because I forgot the harmonic minor scale was an option in Hookpad.