This lesson is all about ways in Japanese to give an excuse/reason in ways we have yet to see thus far in the curriculum.
Also read as しょい, せい means "since/because of/due to" an is essentially the same thing as ため except that せい is used in a sense of blame. The particle that follows is で.
If you use this pattern with something that is generally perceived as being a good thing, it will sound strange and usually unnatural. You would have to do some explaining. The way to express this in a positive fashion is by using のおかげ（さま）で, which doesn't always have to be used in a positive situation. The point is that at least it can.
I was scolded by my dad because of my little sister.
Don't blame me.
We got into a food shortage due to the tsunami.
I didn't come to the party due to an illness.
Although I'm on a break, due to strong winds and pollen, I can't leave the house.
6b. 彼が失敗したのは怠けたからだ。（More natural)
He failed because of his laziness.
I arrived later due to the rain.
He is less sleepy than tired due to his fatigue.
As the adverbial particle か shows uncertainty, it shows in translation when used with せい.
I feel bad because of the uncertainty of the weather.
Mr. Yamada isn't well today apparently because he did bad on his exam.
Kobayashi: You look a little bad; what's wrong?
Deguchi: Yeah, I think my stomache's hurting because of food poisoning.
Nakada: What's wrong?
Kawasaki: My head is hurting cause I might have gotten a cold last night.
Nakada: Well, how about getting home quickly and sleep?
The 一段 verb 託ける means "to use as an excuse" and ～に託けて is often translated as "under (the excuse of)".
The Spanish Conquistadors attacked and utterly destroyed the Aztec Empire under the name of religion.
He skipped work under the excuse of snow.