There are a lot of slightly different definitions, but resilience is essentially a system’s ability to absorb change and maintain itself despite disturbances. This relates to renewable natural resources because all resources rely on some sort of system to maintain themselves.
For example, crops rely on the soil for nutrients, and the soil needs certain fungi to help cycle the nutrients from organic matter back into the soil. If the fungi are removed, nutrients will enter the soil more slowly. When the crop is harvested and the remaining plant material is left to decompose, it will do so more slowly. This has created a less resilient system, and later crop yields will be lower.
Sources:
http://www.ecologyandsociety.org/vol12/iss1/art23/table1.html