“be good at”后面加动词要加ing吗？为什么？

应该跟动词ing形式。

at 是介词，介词后面要跟名词或动名词即动词的ing形式

be good at doing sth 擅长做某事。

例如：He is good at swimming. 他擅长游泳。

动词的-ing形式也是一种非谓语动词。-ing形式仍保留有动词的特征，可以带有其所需要的宾语或状语而构成-ing短语。

-ing的形式：-ing有一般式和完成式。及物动词的-ing还有主动语态和被动语态，而不及物动词的-ing则没有被动语态。

-ing形式的基本用法。
（1）作主语：Seeing is believing.百闻不如一见。Talking is easier than doing. -ing作主语时，如果其结构较长，可用it作形式主语，而将作主语的-ing后置。如：It isn't much good writing to them again. It's no use waiting here.
（2）作表语：Her job is washing and cooking. My hobby is collecting stamps.
（3）作宾语：①作及物动词的宾语。She likes drawing very much.；②作某些短语动词的宾语。
（4）作定语：The sleeping child is only five years old. Do you know the man standing at the gate? 注：-ing形式作定语用时，如果-ing只是一个单词，就位于其修饰的名词之前，如果是-ing短语，就位于其修饰的名词之后，-ing作定语时，被-ing所修饰的名词就是该-ing的逻辑主语。另外，-ing作定语用时，其动作和句子谓语动词所表示的动作是同时进行的，如果不是同时进行的，就不能用-ing作定语，要使用定语从句，如：The girl who wrote a letter there yesterday can speak English very well.
（5）作宾语补足语：We can see steam rising from the wet clothes. 注：当-ing在复合宾语中作宾语补足语用时，句中宾语就是这个-ing的逻辑主语，可以带有这种复合宾语的动词有see, watch, hear, observe, feel, find, have, keep等。
（6）作状语：①时间状语：Seeing Tom, I couldn't help thinking of his brother. 分词在句中作时间状语时，其前一般可加when或while，如：When crossing street, you must be careful. ②原因状语：Being ill, he didn't go to school yesterday. ③方式或伴随状语：Mary stood at the school gate waiting for Betty.