This means to show or prove that somethingis true.

He has weathered plenty of adversity, as the long list of his opponents attests. Media reports in August about moves to oust him came to nothing.

This can also be written as:

As the long list of his opponents attests, he has weathered plenty of adversity.

This can also be written as:

The long list of his opponents attests that he has weathered plenty of adversity.

as…attest… = for example/for instance

• Air pollution can have a negative impact on people’s health, as the rise in respiratory disease in areas with poor air quality attests. (= for example, people living in areas with poor air quality are prone to respiratory disease.)
• Air pollution can have a negative impact on people’s health, as is attested by the rise in respiratory disease in areas with poor air quality.