So I understand that objects in free fall are in an inertial frame, at rest in terms of relativity. However, from a person on the surface of earth, a falling apple is accelerating constantly until it hits the ground.
This is coordinate acceleration, but I still don't understand why it's there: what about the gravitational field makes the movement along the geodesic appear as acceleration to the observer?
Why isn't it, say, a constant velocity or something?
The acceleration on a particle following a geodesic is defined by the Christoffel symbols which are in turn defined in terms of the metric.
More properly, all inertially-moving objects not affected by external forces (including ones affected by gravity - remember, gravity isn't a force in GR) follow geodesics. A geodesic is defined by an initial position and velocity, and from there the geodesic equation (see the wikipedia link) defines how the velocity changes over time (and of course velocity integration gives displacement from the initial position). This results in apparent gravitational acceleration.
It looks like an apple is accelerating at a uniform rate when you drop it because you aren't following a geodesic, because a force is being impressed upon you: when you stand on the ground, the Earth exerts a force on you to keep you from falling into it, which doesn't happen frequently$^\text{citation needed}$. You, held in place by this reaction force, then observe the apple, which is following a geodesic, to accelerate towards the Earth.
The reason that things can start at rest - like an apple - and then begin to accelerate is because geodesics are four-dimensional. If you are "at rest" in a given frame, then your velocity isn't zero, it's just all in the "time direction". In fact, your velocity is actually always equal to exactly $c$. When you accelerate in any spatial direction, your four-velocity doesn't change in magnitude, it just rotates away from the time direction, sort of. Same reason why things start at rest and then fall: geodesics in curved spacetime temporarily curve away from the time-direction and towards the spatial directions, resulting in an apparent acceleration.
(Of course, this is all highly-simplified and is difficult to explain all the way mathematically, but in concept it's not wrong.)
