I understand the idea that, when spacetime is warped by a large mass, objects that try to travel in straight lines instead move along curved geodesics, something Newton would describe as an accelerating force acting on the object in some direction other than the direction of motion, which makes it deviate from its path. That part I get. What I don't understand is how that carries over for stationary objects. If I throw a ball across the room, I can see its path curving as it moves through curved spacetime, but if I hold it in my hand, I can still feel its "weight" pulling down, even though it's not moving along any geodesic. Where does this "force" come from?
A "fictitious force" occurs when your coordinate system is not "natural," in the sense that it is not the right sort of coordinate system to express how Nature works. Basically, if your coordinate system is "simple enough" then you can pretend that your "unnatural" system is really a "natural" one but everything experiences a mysterious "force" which seems to come from nowhere.
In classical mechanics, we already have a great example: a rotating reference frame is not "natural" and therefore has two fictitious forces --centrifugal and Coriolis -- which we can invent. If we invent those then we can pretend that things are standing still! Perversely, the geocentrists were right: you can by all means put the Earth at the center of the Solar system, if you invent the right centrifugal and Coriolis forces to account for its orbit about the Sun. Their only problem was to describe these with "epicycles" -- circles upon circles -- but actually that is now a trick we use all the time, we just call it "Fourier decomposition." (And in any case it's no more physically complicated than a satellite orbiting a Lagrange point.)
Another great example of this is when your car brakes sharply and you seem to be flung "forward": this force is a fictitious force created by the "unnaturalness" of the car's braking motion. Or if you're on a train or subway and you feel suddenly tossed to one side or another, it's hard to remember that actually you are standing still and it is the floor and walls which suddenly tossed themselves at you! We're built to think of the train-car as a "room" and to locate ourselves within it, and if that's our reference frame then our "natural" motion will appear to be acted on by these fictitious forces tossing us side-to-side.
In general relativity, Nature wants to guide particles along geodesics. This means that the appropriate coordinates for describing Nature are coordinates which are in free-fall. Your coordinates are not in free-fall, so they are therefore "unnatural". If you are in free-fall, then the ball in your hand does not feel like it has any weight at all! However your unnatural coordinates are very closely related to the "natural" coordinates that Nature uses, so that you can easily pretend that your coordinates are "natural" too, if you just invent a fictitious force that's pulling everything down. This fictitious force is the gravitational force.
If the ball is in free fall, it moves along a geodesic. In order to deflect it from its geodesic, you must accelerate it by applying a force to it. For instance, you can push it with your hand. If you do so, you will feel a reaction, which you call "the ball's weight".
You write that "if you hold it in your hand, you can still feel its weight". You may call it semantics, but I wouldn't use the word "still", since you can't feel its weight as it flies across the room. When you throw it, its weightless.
The geodesics that objects in free fall follow are actually as straight as possible, not "curved geodesics". It's spacetime that's curved, such that geodesics that are as straight as possible locally don't behave as if they were straight globally.
If you hold a ball in your hand, the ball isn't following a geodesic, i.e., the ball isn't stationary in any inertial frame of reference. The force on the ball is coming from your hand. In terms of proper acceleration, the Earth's surface accelerates upward, which makes you accelerate upward, which makes your hand exert an upward force on the ball.
It may also be helpful to read this answer, which explains more about the Earth's surface accelerating upward.