The truth of the matter was that I refrained from pushing the cat off because I was afraid she `d rip my suit or scratch my eyes out.
这句中,the truth of the matter是主语,was是谓语,that从句是表语.that从句中,I refrained from pushing the cat off后面接一个because的原因状语从句.because的原因状语从句中,有一个省略that的宾语从句：I was afraid (that) she'd rip my suit or scratch my eyes out. 其中she'd的完整形式是 she would . would 是will的过去式. 宾语从句中两个动词词组用or连接,表选择.
翻译是“事实上,我忍住了没有把这只大猫推开,因为我怕她把我衣服撕坏或者把我的眼珠子挖出来.”
However, I replied, “Yes, ma`am, Tiger obviously has some kind of instinct that enables her to accurately judge humans. She sure is a good judge of character. ”
双引号的内容是直接引用当时的原话.里面有个that引导的定语从句,修饰instinct该词.翻译为能使她准确判断人类的直觉.
Even though Tiger happened to be right in my case. I personally put very little faith in an animal’s instinctual ability to judge people.
这两句连起来的话是个让步状语从句,自己翻译吧.
I’d say his reactions have more to do with odors and body movement
I'd say 后面是省略that的宾语从句.