Saying why gets tricky beyond "because of Jupiter", but to clarify on the quote, the statement "Earth's eccentricity follows a 100,000 year cycle" is loosely true but it's also an oversimplification. From Wikipedia.
The Earth's orbit approximates an ellipse. Eccentricity measures the
departure of this ellipse from circularity. The shape of the Earth's
orbit varies between nearly circular (with the lowest eccentricity of
0.000055) and mildly elliptical (highest eccentricity of 0.0679)2 Its geometric or logarithmic mean is 0.0019. The major component of
these variations occurs with a period of 413,000 years (eccentricity
variation of ±0.012). Other components have 95,000-year and
125,000-year cycles (with a beat period of 400,000 years). They
loosely combine into a 100,000-year cycle (variation of −0.03 to +0.02).
If the highest is 0.0679 and the variations are 0.012 and up to 0.03, that's over 50% variation peak to peak. That's amplitude, not period, but if we look at the chart it's clearly not a neat and tidy cycle though it's somewhat close to a 100,000 year period (top line on the chart below).
Source of image.
The cause of Earth's eccentricity variation is the other planets, primarily Jupiter, with Venus and perhaps Mars (small but close) and Saturn also having effects. Whether a particular planet is responsible for a particular period is a good question. I'm not sure.
Ultimately this is a version of the 3 body problem or, more specifically, orbital perturbation, which is complicated mathematics. Unfortunately, I can't give a good explanation of precisely why the gravitational perturbations work out to roughly 100,000 years or, more specifically, 413,000, 95,000 and 125,000. Maybe someone else can.
What would be the simplest set of conditions and/or fewest body system
that would lead to this particular type of periodic variation in
The fewest bodies would be 3. Sun-Earth-Moon is one example. The Moon undergoes eccentricity variation on a much faster cycle than any of the planets. Sun, Earth, Jupiter or any set of Sun and two planets would work too provided the planets were sufficiently large and/or sufficiently close to affect each other's orbits and (I think) you'd want the two planets to not be in orbital resonance. That might create a different kind of pattern.