The term "debt spreads widen" refers to a credit spread that is increasing. This concept is mentioned to explain a spread in yield between corporate bonds and government bonds.
The yield of bonds is based largely on the risk factors associated with the bonds. Investors purchasing corporate bonds assume a higher risk than do investors buying government bonds. Corporations can default on repaying their debts; the government does not.
The difference in yields of corporate bonds and government bonds is a credit spread. Corporate bonds always have a higher yield in general than government bonds. When the yield of corporate bonds keeps increasing while government bond yield remains the same, the debt spread widens, or the credit spread increases.
When the credit spread rises, it is a sign that investors are willing to take on increased or greater risks. When the debt spread shrinks, it shows that investors are being more careful about where they put their money.