They write letters to their favorites,giving them advice and

翻译
他们给他们的偶像们写信,说出自己的看法,提出自己的意见,有时候还模仿偶像们的穿着和举止.
giving them advice and offering their own opinions是做伴随状语的,理解的话,可以理解为：They write letters to their favorites and they give them...
[[[[[[前后主语一致的情况下：用现在分词引导]]]]]
当状语表达原因,主句表达结果时,状语部分放在前面
比如,Having done enough exercise,he got a good score in the exam.
没有任何因果关系,只是表示承接,先后的关系时,状语放在主句后面
比如,I went into the classroom,finding that the class has begun.
(这里,你理解的时候就可以理解为：I went into the classroom and found that the class has begun.
,这和你开始的那个例句的道理一样)
[[[[[[前后主语不一致时,过去分词引导]]]]]
当状语表达原因,主句表达结果时,状语部分放在前面
比如：Blamed by his father,his face turned red.
没有任何因果关系,只是表示承接,先后的关系时,状语放在主句后面:
比如：
He didn't go to school today,known by everyone.
懂了没?