Explain how Woody ended up in Japan.

The other educator answer here does a good job of explaining what Woody does while he is in Japan and what it means to him personally. Instead of repeating this, let us look at some of the historical background of why Woody would be in Japan as part of an occupying American force.
After the surrender of Japan in August of 1945, the United States led the Allied effort to rebuild the shattered nation. This was also the case in Europe, where it was thought that the broken Axis powers would be susceptible to Soviet influence if their economies were not quickly rebuilt. Under the terms of the peace treaty, Japan's government was also to be restructured and its military dismantled. To achieve this, several divisions of US military personnel would spend the next six years occupying Japan.
Woody was one of the 5,000 Japanese American GIs to participate in the occupation. Many served as interpreters. Woody was still a member of the army after being drafted during the war. Unlike Woody's experience during the war, he serves in an integrated unit alongside non-Japanese American soldiers. It was thought that having Japanese Americans in these units would help build connections between the occupiers and the occupied. For many Nisei like Woody, this served as their first direct experience with their ancestral homeland.

Woody is in Japan as part of the American occupying forces. While stationed there, he makes the decision to go visit his family. This takes place almost a year after the atomic bomb was dropped on Hiroshima. At first, Woody is somewhat reluctant to visit. After all, he's part of what the Japanese consider to be an enemy force. Nonetheless, he goes ahead with the visit and is soon very glad that he did. His family welcome him with open arms and are genuinely pleased to see him.
Woody is thoroughly Westernized, a loyal American citizen, completely comfortable in his own identity. Yet his visit to his Japanese family is very important as it enables him to reconnect to his heritage and gain a greater understanding of the pride of the Wakatsukis, that "ancient, inextinguishable dignity."