This is the 1573rd most frequent Russian word.
свидетельствуют
"Testify" or "indicate."
Here, 'свидетельствуют' is used to indicate that the documents present or serve as evidence for the importance of the discovery.
Документы свидетельствуют о важности этого открытия.
The documents testify to the importance of this discovery.
In this sentence, 'свидетельствуют' means that his words show or suggest a deep understanding of the problem.
Его слова свидетельствуют о глубоком понимании проблемы.
His words indicate a deep understanding of the issue.
Here, 'свидетельствуют' suggests that the facts act as evidence supporting the need for change.