They're saying that in the old system, the receiver of the envelope would pay the fee. You'd take the envelope, pay the postman, and then open the envelope and read the letter. 

But this girl and her boyfriend worked out a scheme where she never had to actually open the letter and therefore never needed to pay for its delivery - she just had to hold the envelope for a second, see what her boyfriend wrote on the outside of it, then give the envelope back to the postman and refuse to pay. She didn't need to keep the envelope or open it.