What is brinkmate?

Answer 1

Brinkmate (必死, hisshi) in shogi is threatmate (詰めろ, tsumero) with no escape, like how checkmate (詰み, tsumi) is check (王手, oute) with no escape.

And just like how check is a threat to capture the king, threatmate is a threat to check the king repeatedly, ending with checkmate.

Note: this should not be confused with a bankrank mate in which the king is in the back row and the pawns block the kings escape.