Although he claims he didn't invent it, the Ramma was the one who popularized hephsed meruba, so I will quote what he has to say about it, found in his forward to Toras Chatas. This is my rough translation of dibur hamaschil והנה.
And behold I will save myself in one respect so that the reader will
not suspect me. For sometimes I will write to be lenient for a large
loss or a poor man for an important item, or for the honor of
Shabbos. The reasoning is for in those places it seems to me to be
completely allowed according to halakha. However, the later
authorities were stringent on the matter, I therefore wrote that in a
situation of impossibility, we can place the matter on it's law. And
so I have found in the early ones and later ones who have done this.
The Mahari Minz in his responsa #16 that a poor man throughout the
week and a rich man on the eve of shabbos are the same, however, the
custom is to tell them the reason (of the leniency) this one because
he is poor and this one for the honor of shabbos, so they will not be
confounded that at times I disallow and at times I allow.
I would like to point out that this is very different than what many people think hephsed meruba means. They equate it with what is classically called ra'uy lismoch alav bishaas hadchak. Which means we disagree with that opinion but since he is a great rabbi, his opinion is worth something as a halakhic decision. This is hardly ever used in classic codified halakhic works. What the Ramma describes as hephsed meruba is the exact opposite. An opinion who we agree with, but for whatever reason decided not to rule with lichatchila.