To be visible from the ground, a satellite (including the ISS) must satisfy several conditions:
- It must be above the horizon
- It must be illuminated by the sun
- The observer's sky must be sufficiently dark to not overwhelm the reflected light from the satellite.
Those three conditions basically mean that with the exception of a few extremely bright Iridium flares, all visual satellite observations must take place in fairly narrow windows before sunrise or after sunset (depending on the satellite's reflectiveness) such that the sun is visible to the satellite but not the observer.
The reason successive passes are clustered as all-morning or all-evening has to do with the inclination of the orbit, its orbital period, and the rate of precession of the orbital plane (which also depends on the inclination). The ISS orbits very nearly divides a 24-hour period, so the timing of overhead passes does not change very much from day to day. The orientation of its orbit does rotate (completing a full rotation over a span of about two months or so), however, so what you end up with is several days of morning passes (with occasional interruptions), followed by several days of no visible passes, followed by several days of evening passes, etc.
This is further complicated by the changing seasons altering how the terminator between illumination and shadow is laid out across the planet.