Yes, many black people have larger lips than those of, say, Caucasians, but not all of them. And I don’t know if you have noticed that black people having large lips also tend to have larger nose, or to be more precise, larger nostrils.
The hypothesis is: because black people mostly reside, at least originally, at relatively warm places, the anatomy of their bodies is more inclined towards the facilitation of heat dispersion. The larger lips and nostrils are more efficient in helping the body cooling down in hot climates: while larger nostrils will of course make an increase in the rate of ventilation easier, larger lips creates larger surface area for heat dispersion. If you consider the fact that people are more likely to breathe through the mouth when exposed to extreme heat, the extra surface area those larger lips create could be crucial.
It is also for this reason that people from tropical areas that are not black also tend to have larger lips and nostrils. If you have ever been to South India, you should notice that those people also have relatively larger lips, even though ethnically they are not black.