My husband, who is not Catholic, tells me the Church used to discourage reading the Bible. Is this true and why?

Reading the Bible is of course to be encouraged. The problem is not reading, it is interpreting. If in the past priests once encouraged the faithful to be cautious in reading the Bible, it was only to protect them from the Protestant tendency of private interpretation, which leads to a lot of divisions. That priests ever did discourage the faithful from reading the Bible is exaggerated in terms of its extent and severity. 
 
However, any such warnings ought to be seen in the light of what private interpretation has wrought: namely some 30,000 different denominations of Protestants all claiming biblical authority for their differing views. 
 
Today, Catholics are strongly encouraged to read and pray with their Bible, but to strive an conform their understanding of the text to Church teaching and norms of Catholic Biblical interpretation articulated in the Catechism.