You were right to eliminate B, C, and D right off the bat for using "like" instead of "such as" - "such as" introduces examples, while "like" creates a comparison. We want to say that China and Japan are two examples of countries that hold stores of US dollars, not that countries that hold stores of US dollars are similar to China and Japan.

The only difference between A and E is "holding" vs. "hold". To figure out which one of these is wrong, let's try to condense the sentence by pulling out "such as China and Japan", since it's extra information:

A. The United States has the largest trade deficit of any country on Earth, other nations holding stores of US dollars that increase each year.
vs.
E. The United States has the largest trade deficit of any country on Earth, other nations hold stores of US dollars that increase each year.

We should notice that in E, both "The United ... Earth" and "other ... year" are independent clauses - they can stand on their own as complete sentences. Two independent clauses cannot be separated by only a comma: the result is a comma splice, which is ungrammatical. So we can rule out E.

In A, "other ... year" is not an independent clause, so it can grammatically be separated from "The United ... Earth" with only a comma. So we pick A.