There are two possible interpretations of the question:
- What is the microscopic origin of the normal force?
- If we accept the existence of contact forces, why would the rolling drum apply a normal force (inwards)?
Since I am not at all equipped to answer the first interpretation, and because the question concerns a particular system, I will answer the second interpretation.
Brief heads up for my answer: I will proceed via logical argument, by stating my assumptions and deducing from them that a normal force must be applied by the drum. However, the assumptions I will make (and not the deduction) are the key to the answer, since the normal force will be (almost) a trivial consequence. Also, my answer is a little verbose, but I feel it is useful to examine the assumptions we make when dealing with mechanical systems.
tl;dr the drum applies a normal force to keep the particle inside.
The notion we have from studying configurations in static equilibrium, is that a normal force is applied only to oppose an applied force, to maintain equilibrium.
The scenario in the question, however, is not one of static equilibrium. This creates a question for us: should there still be a normal force exerted by the drum, even though there is no force directly opposite it?
I am assuming that the drum is a rigid constraint: it does not deform and it does not allow objects inside it to go outside by going through the walls.
I am also assuming that the drum is frictionless. More explicitly, it means that any force that the drum applies cannot have a tangential component, and that if the drum does apply a force, it has to be normal to its surface.
Let's assume then that the drum does not apply any force. If the particle, at the position $\theta$ shown, is given a velocity tangent to the drum or with a component along the outward normal, it is easy to imagine that it will, with the only force on it being gravity, leave the confines of the drum.
But since we have posited that the drum is a rigid contraint, this trajectory is a contradiction. Since by Newton's second law, the trajectory is completely by the forces acting on our particle (along with its initial position and velocity) there must be another force acting on it. Since the drum is the only other object in the system, it must be applying a force on the particle, somehow. Since it is frictionless, it must be applying a normal force.
The normal force here is an example of a constraint force. Perhaps part of your confusion is the lack of a physical reason for constraint forces, the way there is for gravity, and the way a reason can be formulated for statics (constraint forces oppose applied forces).
The reason they seem to be an ad-hoc addition to our system, just something that is there because it agrees with experiment without any fundamental theory that tells us it must be there, is because they are ad-hoc additions to our system because we have prior knowledge of our system's evolution. We know, for example, that a particle rolling on the inside of a drum does not go through the walls of the drum, and the drum does not deform (to an observable extent).
Within the framework of Newtonian mechanics, without detailed models of rigid bodies that tell us how contact forces originate from fundamental forces between elementary particles, constraint forces are simply another ingredient we must add.