What does liberty mean to Black Beauty?

I think that you might be referring to chapter 6 here. Amidst all the comfort and pleasure of Birtwick, there is one thing Black Beauty misses: his liberty. He explains that having to wait and serve a human—even a kind master like his—at all times of the day, week after week, year after year, really could upset a horse. This was in contrast to his earliest years, when he could do as he pleased in the meadow of his master. Now, a few hours every week he would get a taste of that liberty again; their master would let the horses loose and they would gallop and roll around and nibble the grass and stand around the chestnut tree talking to each other.