Here is one example[1]:
"One example, which all Russian people know by heart – Volodya Lenin was playing in the house and accidentally broke crystal graphene (crystal bottle for water or drinks). He was afraid that he would get punished, so he lied that he had nothing to do with this accident. But deep inside he was a good boy, so he felt bad about it. And several days later he told his relatives that it was him who broke the thing."
"Lenin was a Mushroom" was just taking this to an extreme and that's why people in early 90s USSR found it funny.