Bonds issued by sovereign nations are often perceived as safe investments. But over time, countries in difficulty economic situations have needed to restructure its debt structure, or see its national economy collapse. During that process, it must restructure the outstanding debt by offering its old bonds holders new instruments that reflect new financial terms. It is a process that sees the emergence of holdout creditors who refuse the proposed restructuring, posing a problem to the reorganization process.
Hence, the exit consent is a formal agreement that allows a majority group of creditors to change the non-financial terms of the bonds in a way that makes the bonds effectively worthless for the minority holdouts. The idea is that creditors willing to restructure can outmaneuver holdouts by using the supermajority voting features of existing bonds to secure changes, which reduce their value as they are tendered in exchange for restructured debt.
Therefore, the threat of an exit consent is used to encourage (or coerce) minority creditors to accept the exchange offer so they are not left with diminished bonds.