This is the 6163rd most frequent Russian word.
тайнами
The word "тайнами" is the instrumental plural form of "тайна," meaning "secrets" or "mysteries" in English.
The word 'тайнами' here is used as the instrumental plural form of 'тайна', meaning 'secrets', and indicates that secrets are the subject of admiration.
Я восхищаюсь их тайнами.
I admire their secrets.
In this sentence, 'тайнами' is instrumental plural of 'тайна', showing it as the means by which they surrounded others.
Они окружили нас своими тайнами.
They surrounded us with their secrets.
Here, 'тайнами' is again instrumental plural, used to indicate the content being shared.