In equity swap, payoff is based on performance of the underlying equity asset. (note that sometime equity swaps may have bonds and other asset classes)
Receiver is the party who receives the performance underlying. If the asset prices go up he will profit the increase in price. If asset prices go down, he will pay the difference to other party. Note that, Receiver is exposing himself to the asset (profit and loss) without owning the asset.
Payer is the party who pays the performance of the underlying to other party. If price go up he will pay the increased amount. In return, Payer will get the fixed amount based on agreed rate on the notional. In addition, if asset prices go down he will receive the difference. So, payer’s market risk is fully removed. However, credit risk (counterparty) still exists.