Context (New York Times),
Edakuni Eiichi, the chef, turns out innovative dishes like daikon rolls stuffed with foie gras and sweet potatoes. The set 10-course meal, which changes monthly, is 3,680 yen (about $40 at 91 yen to the dollar).
In this sentence, at 91 yen to the dollar informs the reader that this is the rate used to calculate the conversion of 3,680, i.e. a dollar is worth 91 yen or 91 yen yields one dollar. If you divide 3680 by 91, you get approximately 40.
Here are similar hypothetical examples:
Including exchange rates in monetary conversions is important in any written work, as these rates are not constant.
At 91 yen to the dollar means assuming that 91 yen is equal to a dollar. (3680 yen divided by 91 yen per dollar equals about 40 dollars.)
However, the wording of the explanatory quote above is not as good as the wording in your example; the article's wording is more natural.
Per is also a common word which is used in a similar situation: 91 yen per dollar is the same as 91 yen to the dollar.