Translation:Her husband is Chinese.
In the Pimsleur course they use 先生 (xiānsheng) for husband, is it less common?
I currently live in China, 先生 is definitely used more for "Mr" or "Sir".
For example: 请问先生，。。。
It depends, 先生 can be used for "husband" in formal conversation.
For example, if a woman says "我先生", she definitely means her husband (while "my sir" is certainly a ridiculous translation).
It could mean "teacher" too, which gives an archaic feeling in Mandarin (but not in some topolects).
The audio does not make clear whether or not ta1 is masculine or feminine here. Both 他的丈夫是中国人 and 她的丈夫是中国人 should be accepted.
Note that 夫 fū is the one comprising 功夫 gong1fu1 (Kung Fu), because Gong Fu literally means "husband to work" or, as we prefer to refer to it in Martial Arts, "married to practice". This is how you'll easily remember 夫 fū.