I haven't heard the lecture, so I don't know which angle Professor Hammond is approaching the subject from, when he states that the ideas of Zhu Xi don't become popular until after his death. I would say that Zhu Xi had quite a following during his lifetime, as one of the top Neo-Confucian thinkers of his day. However, it wasn't until after his death that his commentaries on the Confucian Classics became the basis for the civil service examinations in China, at which point his ideas enjoyed a widespread popularity that they had not achieved prior to that point.

Anyway, my suspicions are that Zhu Xi's success is largely attributable to his writing style, and his decision to print so many things. For example, Lu Jiuyuan was a contemporary Neo-Confucian, whose ideas could be said to rival Zhu Xi's. Now, while Lu Jiuyuan did have some of his works printed, his preferred method of conveying ideas seemed to be travelling around and lecturing, while Zhu Xi stuck much more strongly to getting his ideas written down, and published. When he did write things down, he was fairly comprehensive, and his commentaries on the Confucian Classics (which became the basis for the civil service examinations) could even be compared to textbooks, in a way. He gave a great deal of importance to two main concepts, Qi and Li, which almost served to make his philosophy more straightforward, since almost everything could be traced back to these two concepts. So, when you combine all of this with the popularity his ideas already had prior to his death, it becomes a little easier to understand how his ideas became the basis for the examinations.

If you are looking for a place to begin with the study of Neo-Confucianism, start with Peter Bol's Neo-Confucianism in History. If you need more recommendations, or are looking for information on a more specific aspect of Neo-Confucianism, let me know, and I'll recommend some other things.