1.  They decided to accept the award in the knowledge (that) it would further spread their findings.
以下表达结构相同：
in the knowledge that...
in the hope that...
in the belief that...
这类结构用来引导从句，表示主句行为的原因，翻译时加上“因为”。具体分析的话，that 从句是前面名词的同位语从句。注意翻译：
They decided to accept the award in the knowledge it would further spread their findings.
他们决定接受这个奖项，（因为）他们知道它将进一步传播他们的发现。
They contacted me in the hope that I could show them a better way.
他们联系到我，（因为）他们希望我能给他们展示一种更好的方法。
She acted in the belief that she was doing good.
她这么做,（因为）她相信自己是在做好事。

《牛津词典》

2.  What started as an exercise in curiosity of two academics from Melbourne’s Swinburne University has found vibrations cause earthworms to form patterns in the same way water droplets react to vibrations.
What 引导主语从句，且在从句中作主语。
excercise 的意思等于 experiment（实验），of two academics from Melbourne’s Swinburne University为其定语，说明由这两位学者承担的“实验”。in curiosity 是started 的方式状语，把定语和被修饰语 exercise 分隔。