Answer: She wanted to help his career.
It is not completely clear in the play why Hedda burns the manuscript. It is possible that she did it in order to destroy something that both Lovborg and Mrs. Elvsted deeply loved. It is also possible that she believed this action was going to "free" Lovborg. However, the reason she gives to her husband is that she did it in order to help Tesman's career. He is touched by this act, believing her to love him deeply and wanting to tell the whole world about this act of sacrifice.