The monthly variations are caused by the change in declination of the Moon as it travels (roughly) along the ecliptic. This is caused by the tilt of the Earth's rotation axis, which results in an angle of about 23° between the equator (declination = 0°) and the ecliptic. Around the constellations of Taurus and Gemini the ecliptic lies at much higher declination than near Saggitarius or Ophiuchus. The same effect causes the change in rise and set times of the Sun during a year.
When you're in the northern hemisphere, a higher declination means that the transit altitude of the Moon or Sun is higher. In fact, their whole daily (apparent - caused by the Earth's rotation) paths lie closer to the North star. Because of that, the points where a daily path intersects with your local horizon lie further north, which you see as a change in rise/set azimuth of the Moon.
All this is true even without the 5° inclination of the Moon's orbit w.r.t. the ecliptic. That inclination only mitigates the effect; the Moons declination does not exactly vary between -23 and +23° every month, but these extreme values themselves can lie between +/-18° or +/-28°, depending on whether the inclination effect increases (23+5°) or decreases (23-5°) the effect of the motion along the ecliptic.
Lastly, the rise/set azimuths do not correlate to lunar phase, as do the rise/set times. Both phases and rise/set times are defined based on the Sun's position; we roughly define "noon" as the moment when the Sun is transiting in the south, and call the Moon "New" when it's in the same direction as the Sun, so there's no wonder that the New Moon transits around noon. This is not true for the effects described above, and hence the corresponding periods are different. The mean lunar period between twice the same equinox point (e.g. Aries) is called the Tropical month and lasts ~27.3d (this is roughly the same as the siderial month, but influenced by precession). The effect of the inclination of the Moon's orbit has a mean period of one Draconic month, with a period of ~27.2d. Instead, the Synodic month, the mean period between two cases of New Moon, lasts 29.5d, about 2.2d longer. Hence, the declination and rise/set azimuths of the Moon do not correlate with the phases.