The value of the U.S. dollar, or "greenback," relative to that of other national currencies (e.g., the Japanese yen, euro, British pound) is directly related to the flow of goods and services across borders. As a general principle, the greater a currency's value relative to that of its trade partner, the fewer goods a nation will export to that partner because its goods, being valued in terms of its own currency, are more expensive in the other country's market. Conversely, the weaker the currency relative to that of the trade partner, the more competitive that nation's goods will be, as they will be cheaper in the trade partner's market. When the value of the U.S. dollar rises against the value of the yen, American goods become more expensive for Japanese to purchase, making them less attractive options for those consumers. A weaker yen makes Japanese products cheaper for Americans, precipitating a trade imbalance.
It is because of the correlation between relative values of currencies and trade balances that some countries, most notably China, manipulate the value of their currency to ensure that their products are more affordable for American consumers. The Chinese have long maintained a "peg" between their currency, the renminbi, and the dollar, which ensured that Chinese goods would always be cheaper than American products. This currency peg has been a major irritant in relations between these two massive trade partners, although there has been gradual progress over the past decade in convincing the Chinese to loosen government controls on the renminbi.
In short, a weak or cheap dollar relative to the currencies of trade partners benefits American exporters by making their products less expensive in foreign markets.