I've wondered about this myself before. So, just to elaborate on lordjackes's point, i think the tactic is to get White's lance [香車] on 1c and White's pawn [歩] on 1d so that the 1d-pawn blocks the lance from moving up further. This configuration allows Black to attack the lance on 1c with their right knight [桂馬].
So:
☗P-1f [1六歩] ... ☗P-1e [1五歩] ... ☗P-3f [3六歩] ... ☗N-3g [3七桂] ... ☗P-1d [1四歩],
then (assuming starting with ☖P1c [1三歩]):
☖Px1d [1四歩] ☗P*1c [1三歩] ☖Lx1c [1三香] ☗N-2e [2五桂]
(And, of course, even if you don't attack 1c with the knight, there's still lordjackes's point about the king being blocked from escaping – White would need to move two pieces in order for the king to travel up the first file.)
This knight attack doesn't work if White moves the pawn up.
So: ☗P-1f [1六歩] ☖P-1d [1四歩] and
☗P-3f [3六歩] ... ☗N-3g [3七桂]... ☗P-1e [1五歩] ☖Px1e [1五歩] ☗P*1c [1三歩] ☖Lx1c [1三香] ☗N-2e [2五桂],
then, White just moves:
☖L-1d [1四香]
to avoid capture (as there's no pawn on 1d [1四])
Anyway, that's one explanation i've seen before.
It also allows White's bishop [角] to peep around the corner at ☖B-1c [1三角] defending 2d [2四]. If Black tries to do a Tsukada Special attack without thinking carefully, White's bishop may be able to attack the rook [飛] on 2d and then promote the bishop on 5g (i.e., ☖Bx5g+ [5七角成]).
There are of course variations where White doesn't move to 1d as well as variations where White moves to 1d even though Black doesn't move to 1f.
It's not like white "should" respond with moving the same pawn but it can be useful in the future. Black can play next P-1e so he is gaining more and more space. Even in the future, it could be possible for black to play P-1d and when white takes on Px1d put a pawn on 1c.
If the kings move to that side is good to have some escape route.