What does "in no small way" mean?
"This fact is due in no small way to his talent."
This is a classic example of "litotes" or understatement.
It is an effect very often obtained by using some form of 'double negative'.
In the case of your example we have the expression "no small" which is a kind of double negative: if it is "not small" then it is "big". So, what we have is "in a big way".
The result is that "this fact is due in a big way to his talent", or in other words, "it is his talent that is the major factor.