I managed to have an insight into main facts.

我设法了解了主要事实。

He's claiming to have an insight into how things cause other things.

他的论断洞悉到了事物之间是如何导致成因的。

To have an insight into family life in China and experience the warmth and generosity of that particular family was something I will always treasure.

能够在一个中国家庭里仔细观察，体验这个家庭的热情，慷慨，它将会是我永远珍藏的东西。

The Medicine of Buddhist have an insight into the causation, through repentance, blessing and Great Mercy water, can solve all the complex diseases quickly and effectively.

佛家医学洞察因果，通过忏悔感恩并加持大悲水，可以解决一切疑难杂症，快速有效。

Try to sit in on related classes at your target school. This will enable you to become familiar with your potential professors and also have an insight into the key parts of your target course(s).

最好去想要报考的学校上相关的课程，这样既可以熟悉一下未来的教授，也能对将来要学课程的核心内容有更深刻的认识。