There's interesting information provided in this website, which provides some (presumably) anecdotal justification for the rule.
The first story from the Sarvāstivāda Vinaya Suttavibhaṅga is as follows:
(Thullananda is a Bhikkhuni)
The lay woman went to Thullanandā and said: ‘Are you aware that was
the Elder Mahākassapa, the Buddha’s great disciple, who is greatly
revered by the deities as a virtuous field of merit? If you were to
rise and greet him, what harm would come of that?’
Thullanandā said: ‘Mahākassapa was originally practicing another
religion, [i.e.] Brahmanism. You greatly reverence that, but I do not
respect it.’
The lay woman was annoyed and scolded: ‘These bhikkhunis say, “If you
do what is good you will get merit”, but when they see bhikkhus coming
they do not rise, as if they were women from another religion.’
When the bhikkhunis of few wishes, contented, keepers of ascetic
practices heard about this they were not pleased. They went to the
Buddha and told him everything. For that reason the Buddha summoned
the two-fold Sangha together.
Knowing, he asked: ‘Is it true that you did that thing, or not?’
She answered: ‘It is true, Blessed One.’
The Buddha for this reason in many ways scolded: ‘How can this
bhikkhuni see a monk coming and not rise?’ Having in many ways scolded
for that reason, he said to the bhikkhus: ‘For the sake of ten
benefits, I lay down this precept for bhikkhunis. From today onwards
that precept should be taught:
‘Should a bhikkhuni, seeing a bhikkhu coming, not rise, this is an
offense of pācittiya.’
The second story from the Vinaya of the Mahīśāsakas:
Now at that time bhikkhunis did not bow to monks, did not greet them,
did not receive them, did not invite them to a seat. The bhikkhus were
annoyed, and did not return to teach. Then the bhikkhunis were
foolish, without knowledge, and not able to train in the precepts. The
senior bhikkhunis saw this, looked down on it, and scolded in many
ways. The matter was therefore told to the Buddha. For that reason the
Buddha summoned together the two-fold Sangha.
He asked the bhikkhunis: ‘Is this true or not?’
They replied: ‘It is true, Blessed One.’
The Buddha in many ways scolded them: ‘Did I not already teach the
eight garudhammas as suitable etiquette regarding bhikkhus? From today
onwards, that precept should be thus recited:
‘Should a bhikkhuni, seeing a bhikkhu, not rise up, bow down, and
invite him to a seat, this is an offense of pācittiya.’
For trainees and novices, it is an offense of wrong-doing. If sick, if
previously there is anger and suspicion, with no shared speech
[recitation?], there is no offense.’
And Pācittiya means:
‘Pācittiya’ means: burn, boil, smear, obstruct. If not confessed, it
will obstruct the path. This is the offense: if a bhikkhuni sees a
bhikkhu and does not rise, this is a pācittiya; straightaway seeing
and not rising, straightaway at that point there is pācittiya.