How does keeping the value of the dollar by foreign US bondholders makes their exports more affordable? How does this work?


  In February 2018, China owned \$1.18 trillion of U.S. debt. It's the
  largest foreign holder of U.S. Treasury securities. The second largest
  holder is Japan at \$1.06 trillion. Both Japan and China want to keep
  the value of the dollar higher than the value of their currencies.
  That helps keep their exports affordable for the United States, which
  helps their economies grow. That's why, despite China's occasional
  threats to sell its holdings, both countries are happy to be America's
  biggest foreign bankers. China replaced the United Kingdom as the
  second largest foreign holder on May 31, 2007. That's when it
  increased its holdings to \$699 billion, outpacing the United Kingdom's
  $640 billion.


Source: 
Article - Who Owns US National Debt
Suppose a camera is made in Japan and sold in the US. If the maker of the camera wants 10k Yen for it, and the ultimate buyer is paying dollars, then somewhere along the line, dollars have to be exchanged for dollars. For instance, the company that imports the camera might go out and buy 10k Yen, then give the Yen to the company that made the camera. They then pass on the cost of buying the Yen to the camera store, who passes it on to the customer. The dollar being strong with respect to the Yen means that one dollar can buy a lot of Yen. So that means that it doesn't cost very much dollars to buy the 10k Yen, which means that the camera store in the US can sell the camera for fewer dollars, which means that people are more willing to buy the camera, which means that the camera manufacturer gets more business. 

You're also asking how the strength of the dollar is affected. It's a matter of supply and demand: every time Americans buy stuff from other countries, they are supplying dollars, and every time they sell things to other countries, they are demanding dollars. The strength of the dollar depends heavily then on how much the US exports (which strengthens the dollar) versus imports (which weakens the dollar). Since transactions between the US and other countries involve trading goods and dollars, the effect on the dollar is opposite of how the goods are moving from the US perspective: when someone in the US sells a good, that can be viewed as "buying" dollars, and the US buying goods can be viewed as "selling" dollars. A strong (or "expensive") dollar means that US goods are also expensive, but foreign goods are cheap, and a weak dollar means that US goods are cheap, but foreign goods are expensive. 

Companies that facilitate exports generally benefit from a weaker dollar (the goods are cheap, so it's easier to sell them). This includes export companies, and people who work in industries whose markets are largely overseas.  Companies that facilitate imports generally benefit from a stronger dollar. This includes import companies, and companies that that sell a lot of foreign-sourced goods, which in the camera example would include the camera store.

Bonds aren't generally included when calculating exports, but they do have largely the same effect: US treasury bonds are denominated in dollars, so if people in other countries want to buy them, they have to buy dollars, which drives up the value of the dollar.   You can think of it in terms of analogy with a household: exports are like people going out and working for a paycheck, while imports are like buying stuff. A bond is like getting a loan, such as a mortgage or a car loan. The more money someone makes at their job, and the more loans they get, the more the cash they have to spend. A dealership wants to help someone get a car loan, because that makes it easier to but a car. Similarly, China and Japan want the US to be able to sell their bonds, because that makes it easier to buy stuff from China and Japan. Whenever China or Japan buy a treasury bond, they give more money to the US, and the US then uses that money to buy stuff. If China were to sell off their bonds, that would make it harder for the US to sell new bonds, which would make it harder for Americans to buy stuff from China.