It's not difficult to show that anything moving in a constant speed in a circular path has an acceleration (rate of change of the velocity vector) towards the centre of the circle. The objects in the washing machine drum have just such an acceleration. But according to $\vec{F}=m \vec{a}$ they must experience a force towards the centre of the circle in order to have this acceleration. The force comes from the walls of the washing-machine drum, pushing inwards on them. The objects therefore exert an 'outward' force on the drum. This a case of Newton's third law: if body A exerts a force on body B, then B exerts an equal and opposite force on A. If you exert a force on a chair by leaning on it, the chair exerts a force in the other direction on you – which probably stops you from falling over! There's nothing 'apparent' or 'fictitious' about the (inward) force the drum exerts on the objects in it, or the (outward) force that the objects exert on the drum.

If you want to, you can call the force that enables a body moving in a circle to have the compulsory acceleration towards the centre of the circle the *centripetal force*. In this case it comes from the walls of the washing machine drum pushing inwards. 

We've been looking at what's going on as 'outside observers', in a so-called *inertial frame of reference*, in which the drum is turning round. An inertial frame of reference is one in which Newton's laws apply. Indeed we applied them in the first paragraph (above). If we look at things from a reference frame moving round with the objects in the drum, then in this frame the objects aren't accelerating but do have a so-called centrifugal force acting *outwards* on them. It's called a fictitious force, because, among other things, there's no external body that gives rise to it. Newton's laws don't apply in this accelerating frame of reference. But in this frame of reference there's still an inward force force from the drum on the object, and if *you* are the object in a large drum, you feel a real (inward) force from the drum, which you may well think of as balancing an outward centrifugal force acting on you. I'd argue that the force you *feel* is the inward force from the drum... 

If you're just beginning a serious study of Physics, it's much better to try and understand the explanation I gave in the first paragraph (that is to understand Physics in an inertial frame of reference where Newton's laws apply) than to concern yourself with so-called fictitious force in non-inertial frames.