The mysterious role of 占 in 对占人口绝大多数的民众来说，还是觉得有钱应该省着花
占人口绝大多数的 is a compound adjective for 民众. 人口 means population, noun. 绝大多数的 means most, adjective.



If you take away 占, you got to take away 人口 as well to keep it grammatically sound. If you remove 占 and leave 人口 there, it's like this English sentence, "For the people that most part of population, money should be spent wise." you feel a bit strange because something is missing in between "that" and "most".
That's easy, look at this:


  占全年国内消费水平很大比例的是节假日的消费 (Simplified)
  
  占全年國內消費水平很大比例的是節假日的消費 (Traditional)
  
  A very large portion of the national consumption is from that in holiday. (One possible English translation)

As for 省着花, 花 really mean to spend money. 省着, an adverb for 花, means "spend money for those necessary things only". The pattern of "verb + 着", also shown in another post, can be an adverb (as in 省着花) or indicating action in process as in 他正弹着吉他, tā zhèng tán zhe jí ta, "He's playing guitar now."


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