These fragments will be further amplified during subsequent rounds at lower temperatures, and will out compete the nonspecific sequences to which the primers may bind at those lower temperatures. If the primer initially (during the higher-temperature phases) binds to the sequence of interest, subsequent rounds of polymerase chain reaction can be performed upon the product to further amplify those fragments.
In future cycles, these pieces will be expanded upon at reduced temperatures, and they will outcompete non-specific sequences that primers may bind to at lower temperatures. If a primer initially (during higher-temperature phases) binds to the desired sequence, additional rounds of PCR can be carried out on the resulting product to further amplify those fragments.