The phrase you are asking about can be found in Chapter 7 of this book. Douglass uses it to refer to the ability to read. He talks about how he used to go out among the poor white kids and give them bread that he had (he had more to eat than they did). In return, he would get from them the bread of knowledge as they would help him learn to read.
What Douglass means by this phrase is that knowledge was very important to him. We often use the word bread to simply mean nourishment in general -- something that gives us life. Douglass refers to knowledge in this way because getting it was so important to his life.
In Chapter 7 of The Narrative of the Life of Frederick Douglass, Douglass describes his life in Baltimore as a young slave in the city. His mistress teaches him the elementary steps of reading, as she does not yet realize that the society around her condemns the teaching of slaves. The society believes that reading will corrupt them and make them unfit as slaves.
Douglass writes, "I used also to carry bread with me...I was much better off in this regard than many of the poor white children in our neighborhood. This bread I used to bestow upon the hungry little urchins, who, in return, would give me that more valuable bread of knowledge." As an urban slave, as opposed to a slave on the plantation, he has access to bread, which makes him much better fed than some of the poor whites around him. He uses this bread to bribe white children into teaching him to read. He believes that learning to read is more important than having food; therefore the "bread of knowledge" is more important than actual bread.
Douglass's readers were well acquainted with the Bible, in which bread is often equated with the essence of life. To Douglass, knowledge is the essential part of life, and he is willing to take a risk by bribing white children in order to learn. He knows that learning will help free him from slavery because he will have the ability to free his mind and body from his captors.
Douglass' use of the word "bread" suggests that knowledge is as necessary to human life as is food.
Furthermore, Douglass is clearly playing upon his readers' knowledge of the Bible. (His memoir was intended as an argument against slavery.) In the Bible, references to bread in the New Testament are numerous, and Jesus' body itself is equated with bread, among other things.
The Lord's Prayer is Jesus' own words in which he asks for "our daily bread"--the stuff of life here are again directly equated to our spiritual well-being.
Therefore, bread, and by metaphor, knowledge, take on a sacramental, holy quality for Douglass, and, he hopes, in the minds of his readers.
Knowledge, for Douglass and his audience, is not merely a matter of literacy, it is also about becoming fully human, and fully capable of developing spiritually, morally and emotionally.