The second part of the centaur's Equine Build racial trait says (MP:MotM, p. 9):
In addition, any climb that requires hands and feet is especially difficult for you because of your equine legs. When you make such a climb, each foot of movement costs you 4 extra feet, instead of the normal 1 extra foot.
Can being a ranger completely negate this debuff? The Land's Stride class feature sounds like it could, but I'm unsure.
The ranger’s Land's Stride feature says (emphasis mine):
Starting at 8th level, moving through nonmagical difficult terrain costs you no extra movement.
The extra movement cost for equine build is not due to difficult terrain; it is due to climbing. As such, Land's Stride does not apply to it.
You can also see that climbing is different from difficult terrain by reading the general rules for climbing (PHB, page 182):
Each foot of movement costs 1 extra foot (2 extra feet in difficult terrain) when you’re climbing, swimming, or crawling.
P.S. What happens in the situation where you are climbing in difficult terrain? Would it negate the entire effect then?
I don't think it should, because Equine Build refers to "the normal 1 extra foot" that is caused by climbing. It seems this was carefully worded to not say "for each extra foot". In this case, as a centaur ranger you would spend 4 extra feet per foot when climbing. Then if you have it, you apply Land's Stride to avoid the added effects of non-magical difficult terrain (which otherwise would cost you another extra foot per original foot of climbing).
I think the root of this question is nothing to do with either centaurs or rangers, but a fundamental misunderstanding regarding the climb mechanic in D&D 5e.
As the basic rules go, climbing costs 1 extra foot of movement for each foot moved. Difficult terrain also costs 1 extra foot of movement of each foot moved. But climbing is not difficult terrain; they just have similar mechanical effects.
An ability that allows you to ignore difficult terrain, such as Land's Stride, does not allow you to ignore the extra movement cost of climbing or swimming. Similarly, climb or swim speeds let you ignore the extra movement cost for those kinds of movement, but don't let you ignore difficult terrain. So a DM could decide that climbing a crumbly cliff face counts as moving through difficult terrain, and thus costs triple to move across (unless you have a climb speed, or a way to ignore difficult terrain, or both, each of which would reduce the cost by 1).
So Land's Stride doesn't negate the Centaur's unique difficulty with climbing, because Land's Stride wasn't helping with climbing in the first place. They're just two different things.
Equine Build:
In addition, any climb that requires hands and feet is especially difficult for you because of your equine legs. When you make such a climb, each foot of movement costs you 4 extra feet, instead of the normal 1 extra foot.
Land Stride:
Starting at 6th level, moving through nonmagical difficult terrain costs you no extra movement. You can also pass through nonmagical plants without being slowed by them and without taking damage from them if they have thorns, spines, or a similar hazard.
One way to look at it is that, when you climb, the only thing getting in your way is your own legs. Land stride won't help you with that.
