There is no outward force on the merry-go-round. The force is inward.
I'll start with linear motion, then go to rotational motion. If you have a powerful car, you have felt yourself sink back into the seat and then felt that push on your back when you step on the accelerator. You sink back into the seat because of Newton's first law. When you first step on the accelerator, your velocity is not quite the same as the car's. Newton's first law says you'll keep going with that velocity. You sink into the seat. The car can push against you once the seat cushion is compressed. The car is accelerating forwards, and yet you feel a force acting from behind you.
The same thing happens when you step on the brakes, hard. Your velocity takes you forward relative to the car until you seatbelt locks. Now you will feel pressure on your chest pushing you rearward. Once again the force comes from a point opposite the direction of the acceleration.
The exact same thing happens once again when you take a hard turn toward the passenger seat (that's a hard right turn in the US, a hard left turn in Europe). Just as before, you keep your current velocity until a force acts on you to change your velocity. When you take that corner, the difference between your velocity and the car's moves you toward the car door. Now the door pushes on you. Just as before, the push comes from the opposite side of the direction of the acceleration. You are accelerating toward the passenger seat. That's toward the center of the curvature of the turn. There is no outward force.
I suspect the issue that gets people so confused about merry-go-rounds and the like is a misunderstanding of what acceleration is. Any change in velocity involves an acceleration. Velocity has a magnitude (speed) and a direction. Changing the speed of a car requires some force. Changing the direction in which a car is moving also requires a force. The force is always in the direction of the change in the velocity vector. In the case of a turning car, or a person sitting on a merry-go-round, the change in velocity is toward the center of curvature, or inward.
Comments have suggested I talk about centrifugal force. I don't think that's a good idea. Understanding Newton's laws is a prerequisite for understanding those fictitious forces.
So let's go back to the car. I assume you have no problem understanding that inertia is the reason you sink into the seat a bit after stepping on the accelerator, or that inertia is the reason you move forward a bit with respect to the car after hitting the brakes. Your velocity is slightly different from the car's velocity for a short time after you step on the accelerator or hit the brakes. Your position with respect to the car is the difference between your velocity and the car's, multiplied by the small amount of time.
Suppose you're traveling at 70 kph (43.5 mph) and you hit the accelerator. Assuming a decent but not great acceleration, the car will be going 71 kph a twentieth of a second later. (That corresponds to a 0-60 mph time of 4.8 seconds.) The car's going 71 kph, but you will still be moving at about 70 kph. Multiplying the car's average speed over that interval by that 1/20 second time interval yields 6.9 millimeters (0.27 inches), which is how much you've moved with respect to the car. That's just enough to sink into the seat and feel the acceleration. The same happens when you hit the brakes, only now you move forward initially instead of sinking into the seat. There's no force this forward or backward motion happen. It's just inertia. Your body keeps that 70 kph velocity until a force does act on your body to change that velocity.
Now let's suppose that instead of changing speed, you go around a corner at a constant speed of 70 kph. Suppose you're going straight north at 70 kph at the start of the turn. In that 1/20 second, the car's velocity changes from 70 kph due north to 69.998 kph north plus 0.99997 kph east. Almost all of the change in velocity is orthogonal to your velocity. This means that 1/20 second interval, you have slid sideways in the car by 6.9 millimeters. No force was involved in this sideways slide. It's just momentum, no different than sinking back into the seat when you hit the accelerator or leaning forward when you hit the brakes.