In general, the past perfect is used to indicate that an event took place some time in the past, before some other event in the past. The simple past just indicates that both events are in the past, without specifying their relative times. Thus, you will often find a past perfect combined with a simple past. This means that the action described by the past perfect was completed before the action described by the simple past.
For example, suppose you write
I worked at Foobar Manufacturing and I got a job at Plugh Garage.
As both are in the simple past, this does not tell the reader the relative times. Did you work at Foobar for a while and then later get a job at Plugh Garage? Were you working both jobs at the same time? Maybe Plugh came first.
I had been working at Foobar Manufacturing when I got a job at Plugh Garage.
This clearly tells us that Foobar came first, then Plugh. (It doesn't explicitly say that you quit the job at Foobar when you got the job at Plugh, but most readers would assume that was what you meant unless you specified otherwise.)
All that said, the example you give is just a shade more complex. Sometimes when there are multiple verbs in the past perfect, we omit "had"s after the first. Usually this is done when they come close together. Like:
I had planned and prepared for this for many years before I finally did it.
Both "planned" and "prepared" are in the past perfect. We could just as well have said "I had planned and had prepared", but we leave out the second "had" because it is redundant.
In your example, I believe the writer's intent is that both "had been the worst" and "asked to marry" are past perfect events that precede "he thought what an appalling experience". The writer has omitted the second "had" as redundant.
If you took the sentence out of context and just wrote, "It had been the worst moment of his life when he asked Minta to marry him", grammatically that could be read as placing the "asked" outside of the past perfect, and thus be saying that it was already the worst moment of his life, and then he asked Minta to marry him. The thing that would make that reading unlikely is the word "moment". Logically we think of a moment as only long enough for one event: how much could he do in a moment before asking Minta to marry him? If it said, "It had been the worst DAY of his life when he asked Minta to marry him" -- again, taking that one sentence with no context -- I think the reader would interpret that to mean that his day had gone very badly up to that point, and then he asked Minta to marry him. We would expect the next sentence to say whether she turned him down and made it worse or accepted him and made it all okay.