。
I'd rather die than lose face.
Literal
Honour [object-を] lose [than-より] died [side-ほう] [subject-が] is-better.
A dramatic comparative: X より Y の方がいい ('Y is better than X'). Here replaced with まし instead of いい — まし means 'preferable (as the lesser evil).' 面目を失う ('to lose face') is a calque of the English idiom, well-established in Japanese. The whole sentence is a stock expression of wounded pride or samurai-style sentiment.