Lunar theory is complicated. ;)
There are two main causes for the variation shown in your graph.
Firstly, the Moon's orbital plane is not aligned with the Earth's equator. It's tilted by ~5° to the ecliptic (the Earth's orbital plane), which is itself tilted by ~23° to the equatorial plane.
Secondly, the Moon's orbit is rather elliptical. It has an eccentricity of around 0.055, which is more than 3 times the eccentricity of Earth's orbit. The Moon's orbital speed is fastest when it's closest to Earth, and slowest when it's furthest from Earth.
These two effects combine to vary the Moon's angular speed across the sky relative to the background of stars. This is similar to what happens with the motion of the Sun, but more complicated. The variation in the Sun's speed is known as the Equation of Time; I have some explanation about that in this answer and graphs showing the combined effects of the eccentricity and orbital tilt.
As I said above, with moonrise & moonset, things are more complicated than with sunrise & sunset and the relatively simple Equation of Time. As well as being more eccentric, the Moon's orbit has a couple of short precession cycles.
the orbit of the Moon undergoes two important types of precessional motion: apsidal and nodal.
Apsidal precession means that
the major axis of the Moon's elliptic orbit (the line of the apsides from perigee to apogee), precesses eastward by 360° in approximately 8.85 years.
So the point in its orbit where the Moon is closest to the Earth changes over a 8.85 year cycle.
Another type of lunar orbit precession is that of the plane of the Moon's orbit. The period of the lunar nodal precession is defined as the time it takes the ascending node to move through 360° relative to the vernal equinox (autumnal equinox in Southern Hemisphere). It is about 18.6 years and the direction of motion is westward, i.e. in the direction opposite to the Earth's orbit around the Sun if seen from the celestial north.
In other words, the plane of the Moon's orbit "wobbles" relative to the ecliptic plane. The wobble is fairly regular, but it's affected by the distance to the Sun, and also by the other planets. I have a 3D diagram of this motion in this answer for a very simplified system which just focuses on the nodal precession so it pretends the orbit's perfectly circular and the motion is uniform. There's a related diagram here.
Other related Wikipedia articles that you may find useful include Orbit of the Moon and Saros (astronomy)
Here are some plots produced using data from JPL Horizons.
This plot shows the speed of the right ascension of the Sun for 2022, for each day at 00:00 UTC, relative to the centre of the Earth. The speed is in arc-seconds per hour. This graph is correlated with the variation in the length of the solar day. To compute rising and setting times we also need the declination, and the observer's location.
The speed is fastest near the solstices, and slowest near the equinoxes, due to the obliquity of the ecliptic. It's also fastest near perihelion and slowest near aphelion.
Here's the equivalent plot for the Moon. It combines the effects due to the Moon & Earth's orbit around their barycentre with the effects due to the Earth-Moon system's orbit around the Sun.
Here's the plotting script, running on the SageMathCell server, so you can make your own plots.
Here's a similar plot to the one in the question, produced using Horizons. It shows the rising and setting time differences for 2022, from Sydney Observatory.
According to the Horizons manual, the rising & setting times should be accurate to within two minutes. The calculations include the effects of atmospheric refraction.
Each plotted point has a horizontal coordinate corresponding to the time of the event, with the vertical coordinate giving the time difference from the corresponding event on the previous day. As in the OP plot, the rising time curve is red, the setting time curve is blue.
Here's its plotting script. This script takes a while to run because it computes a year's worth of data with a one minute time step.