Why would a particle in an extra dimension appear not as one particle, but a set of particles?

I was reading an article in this months issue of Physics World magazine on the three main theories of extra dimensions and stumbled across something I didn't quite understand when the author began talking about detecting particles in extra dimensions at a particle lab, such as the LHC in Geneva, Switzerland.

The energy of a particle in 3-dimensional space consists of its rest energy, $E=mc^2$, and the kinetic energy of its motion. If extra dimensions do exist, then the particle will have extra space to move in, so will obtain an additional, independent contribution to its kinetic energy. Since we don't observe the motion of the particle in the extra dimension, this kinetic energy will be interpreted as rest energy, or in other words, the mass of the particle.

This is all understood perfectly fine, but it is this quote that comes next that confuses me:

To us, the particle would not look like one particle, but a set of particles - all with different masses.

Why would the particle look like a set of particles rather than just the one that is being observed?

Furthermore, why would they all have different masses?

How many particles would there be in this set?

Please keep your answers as simple as possible, as I am just a Layman.

It is not so much as one single particle will be seen with different masses as it is that that one type of particle will be seen as having multiple different masses when it is detected multiple times.

For example if the extra dimension is like a rolled up microscopic cylinder, the particle can have an infinite number of discrete masses starting from the mass which has no extra kinetic energy in the rolled up cylindrical direction, to masses that have 1, 2, ... units of momentum in the extra rolled up cylindrical direction. These momenta are quantized since only whole wavelengths of the particles wavefunction are allowed around the cylindrical dimension and each unit of extra momentum around the cylinder will be seen as additional rest mass. These are called the Kaluza-Klein tower of excitations of the basic particle. Kaluza-Klein theory was developed in the 1920s in an attempt to unify gravitation (general relativity) and electromagnetism (Maxwells equations).

By the way, according to this source:

You may also be interested to know that the original 1921 theory has evolved into today's string theory, as both share the idea of using multiple extra space dimensions to describe the world. The most advanced version of strings is known as M-theory, which utilizes an 11-dimensional spacetime having seven compactified extra space dimensions – a far cry from Kaluza's original single extra dimension!

By the way, Kaluza originally came up with his idea in 1919 and communicated it to Einstein in hope that the great scientist would recommend it for publication. But Einstein, who expressed great admiration for Kaluza's idea, sat on it for two years before recommending it. I'm sure this did not sit well with Kaluza

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The key is in the phrase following immediately after

To us, the particle would not look like one particle, but a set of particles – all with different masses.

which is

The faster the particle moves along the extra dimension, the larger this apparent mass seems to be.

(here is a link)

I think that it means that the same particle in extra dimension may manifest as different particles to us. But not all at once. That is, you won't see one particle as being more particles simultaneously. One massless particle will have a momentum in $4+k$ dimensions, and zero rest mass. When projecting the $4+k$ momentum on our spacetime, we will obtain the $4$-momentum. If we take in our spacetime a reference frame in which the projection of the particle is at rest, the projection of the $4+k$ momentum will give the rest mass (as we see it). But this depends on the angle made by the particle in the $4+k$ space with our spacetime.