This is an inflected form of the word mangyari.
To put it in simple terms, you can translate “Mangyaring…” into English as “Please…” It sounds very stilted and old-fashioned though.
Filipinos are more likely to put sentences in plural form rather than use this Mangyaring… construction. An alternative could be to use “Paki…”
Compare the following sentences, which have a rough English translation to approximate the nuances of each.
Mangyaring hintayin mo ako.
If thou couldst wait for me…
Hintayin mo ako.
Wait for me. (bluntly saying to friend or younger person)
Pakihintay mo naman ako.
C’mon, wait for me. (affectionately said to friend)
This is the most ideal when you’re a clerk in an office or a nurse speaking to a patient in a hospital. See another example below of this plural + po construction.
Isulat po ninyo dito.
Please write it here.
In Tagalog, the plural form is used even when speaking to only one person — when the person being spoken to is older.
The following can be used in very formal circumstances:
Mangyaring tumayo tayo.
If we could please stand.
Mangyaring umupo po tayo.
If we could please sit.
Mangyaring tumahimik lang tayo.
Please, let’s just be quiet.
Mangyaring ikabit ang inyong sinturong pangkaligtasan.
Please fasten your seatbelts.