Only 2.4% of the 113th Congresss were Asian-Americans; by one count, fewer than 2% of state legislators are.
What's the meaning of "by one count"?
For many things only estimates of the number of people is possible, for example numbers of people at an unticketed public event, or the number of people who watch a television program. There are several methods of making those estimates. Hence
by one count refers to the number generated by one of the (possibly many) method of estimating the count.
According to Wictionary, the noun count has the following meaning:
The act of counting or tallying a quantity.
Give the chairs a quick count to check if we have enough.
It is not extremely difficult to count how many Asian-Americans are sitting in Congress (2.4%) and 50 states' legislatures (fewer than 2%).
They just counted all of them once and it revealed the fact.