此句中的“on + 动名词”结构表示时间，意思是“当……的时候”“一……就”。又如：

On entering the room, she saw him. 她一进这个房间便看到了他

On receiving the news, she fell into a dead faint. 听到消息，她昏死过去。

On hearing the crash, he rushed out of the house. 他一听到轰隆声，便冲出房子。

你的句子中的动名词为被动式，意思是“当被……的时候”，你的句子意思是：

On being introduced to each other, men should shake hands. 当被人介绍彼此认识时，男人们应该握手。

分析：

状语：On being introduced to each other

主语：men 

谓语：should shake 

宾语：hands 

更多类似例句(动名词为被动式)如：

On being introduced, British people often shake hands. 英国人被介绍给别人的时候常常和对方握手。

On being informed the flight would be delayed, we made other arrangements. 在我们得知飞机要推迟起飞后，我们另作了安排。

On being shown this book, no one would believe that it was written by a ten-year-old girl. 看到这本书，谁都不会相信是一个十岁的女孩写的。