The example they give is about a hypothetical NN used to assess creditworthiness and approve or deny bank loans.
The 'clean' NN would make a reasonable attempt to decide whether or not someone should be offered credit, based on the information available to the bank at the time.
The 'backdoored' NN would, however, be made to unconditionally approve a loan, if some very particular, carefully crafted conditions were met. For example, a loan of $10,000 might be rejected, but a loan to the same individual of precisely $10,004.26 would be approved.
Crucially, the agency that performed the training would be able to know what criteria would result in these 'false' approvals, but it would be mathematically very difficult to detect that they existed without that inside knowledge.