I was scared and feeling pretty anxious,this being my first time in a new country.为什么这里用this being?I hope this book will be of some help to beginners of English.为什么用of some help?Professor Li's book will show you how what you have observed can be used in other context.为什么用how what you have observed 为什么how 和what一起用?The old man was so angry and spoke so fast that none of his children understood what what he said meant?为什么用what what?

第一个的话首先一个句子里只能有一个动词，所以这里的be要用非谓语动词的形式。然后个人觉得是不是和there be句型有关，要不就是这里没有被动关系，所以用-ing表主动。
第二个这个很难解释啊，你可以把be of +n.看做一个固定词组的，就这么记吧!
第三个这个里面套了个从句，Professor Li's book will show you how （what you have observed） can be used in other context.how表示怎样，然后（）里的是一个主语从句，仔细分析一下句型就知道了。
最后一个，也是一个从句套用的。The old man was so angry and spoke so fast that none of his children understood 【what（ what he said） meant?】（）中的是一个主语从句，meant是从句中的谓语动词，然后【】里的是一个宾语从句，调整过来就是what he said meant what
呼呼，累死我了，你就不能把悬赏分抬高点啊！！

将二楼的1.2题解释结合一楼的3.4题解释就完美了。

I was scared and feeling pretty anxious,this being my first time in a new country.为什么这里用this being?
this being 是独立主格结构.请比较下面两句话的不同：
I was scared and feeling pretty anxious,this being my first time in a new country.
I was scared and feeling pretty anxious,and this was my first time in a new country.
I hope this book will be of some help to beginners of English.为什么用of some help?
这里be of help=be helpful
Professor Li's book will show you how what you have observed can be used in other context.为什么用how what you have observed 为什么how 和what一起用?这里不是how和what连用,而是这个句子可以分为几个层次.句子主干Professor Li's book will show you,其后用了一个how引导的从句作宾语从句,而这个how引导的宾语从句的主语又是由what引导的主语从句充当的.
The old man was so angry and spoke so fast that none of his children understood what what he said meant?为什么用what what?
这句中so...that引导了一个结果状语从句,在结果状语从句中understood的宾语是一个what引导的疑问句,而这个疑问句的主语则又是一个what引导的主语从句来充当,意为“他的孩子们没有一个理解他所说的是什么意思”.

第一个是个独立主格结构（主要作状语），说明前面的原因。
第二个是介词of 和抽象名词的搭配：of use=useful等等
第三个就是疑问代词，可以这样连用，句子表示如何观察并且观察什么：How and when do you come？你什么时候，怎么来的？
第四个的第一个What是指老人说话太快，没人知道他说“什么”，而第二个What是指也没人知道老人说的是“什么”意思，因为生气，说话快，所以既不知道说的什么，也不知到什么意思。