我认为，在没有语境的前提下，命题人最好不要出这样的选择题考查学生。除了B项为错误之外，其他答案都正确。
A.   It was at about this time that her mother started to go to the cooking class.
B.  That was about this time her mother started to go to the cooking class.
C.  It was at about this time when her mother started to go to the cooking class.
D.  It was about this time that her mother started to go to the cooking class.
逐个分析如下：
第一句：正确，是典型的强调句。还原后为：At about this time her mother started to go to the cooking class.
第二句：错误，是不合乎语法的结构。
第三句：正确。It 指代“事件”，不是“时间”。was 等于happened。而when her mother started to go to the cooking class. 是 this time 的定语从句。等于说：
Something happened at about this time when her mother started to go to the cooking class.

还有一种分析，是把when her mother started to go to the cooking class 看作并列分句或状语从句。不过，我认为，该从句与this time 存在修饰关系，故分析为“定语从句”。

有网友可能会问：能不能跟A一样，把C也看作强调句呢？答案是否定的。

如果被强调的是时间副词，强调结构除了用 that 外，还可以用 when。如果被强调的时间是介词短语，则不能用 when ，只用 that。例如：　　
It was yesterday that (when) he bought an English book in town.
It was last week that (when) we worked in the countryside.　　
It is at ten o’clock that the meeting will take place.（不能用when）

第四句：正确，跟第一句相同，也是强调句。唯一的区别就是省略了介词 at，而且这种省略很正常。about 用在this time, the time, that time之前时，为了简练一般省略介词 at：  (at) about this time。如果没有about，原则保留 at。否则，可能被误解为：“这一次；那一次”。