tl;dr because the ISS orbit is not equatorial, choosing the right launch time is non-trivial and solutions are only approximate.
ISS launch windows are complicated - you have to match both plane (i.e. the Earth needs to turn so that the Cape/Baikonur is under the ISS's orbit path) and phase (i.e. at the time the plane match happens, the station needs to be at the right part of its orbit for the spacecraft to meet the ISS right after launch).
A launch site crosses the plane of the ISS orbit twice a day, but the phase angle at those crossings is often quite far off of optimal - you want the ISS to be a bit behind at launch so that by the time your spacecraft gets up to the right altitude/speed it's matched up. So you have to find the minimum phase angle at launch window for a targeted several weeks or so, then have the ISS make slight orbit changes to bring the phase angle closer to optimal. Once you're up and have as low of a phase-angle difference as possibly, you then have to spend a bunch of 90-ish-minute orbits getting your position in orbit matched up to the ISS.
At fastest, you can get down to the single-digit-hours range, but that takes a good selection of launch window, a bit of a burn in advance from the ISS to tweak its phasing, tracking systems that don't waste an orbit determining where exactly you are, and no scrubs.
> The long sought attempt at launching a Russian Progress resupply vehicle to the International Space Station and having it rendezvous with the orbital complex after just two and a half orbits and just under 4 hours in space is being attempted.
...
> As with the other fast track rendezvous attempts, the reboost and phasing alignment of the Space Station with the Baikonur Cosmodrome is only available for this launch day. A scrub to today’s instantaneous launch of Progress MS-09 would have resulted in the craft having to fly a two-day, 36 orbit phasing profile to the ISS on a realigned launch date later this week.
If you don't mind me simplifying your answer a bit...
Imagine the orbit of the ISS like a tilted hula hoop around the Earth, with the Earth revolving inside it. Because the Earth rotates once every 24 hours, any given spot on Earth will be directly underneath that hula hoop twice per day (once on either side of the hoop).
When launching to the ISS, you need to launch when that orbit "hoop" is directly overhead. That's the bit that really matters, because you need to be in the same orbit. The ISS itself is almost certainly not overhead (it may be on the other side of the planet entirely), and so it will take some time to catch up to it.
ISS launch windows are complicated - you have to match both plane (i.e. the Earth needs to turn so that the Cape/Baikonur is under the ISS's orbit path) and phase (i.e. at the time the plane match happens, the station needs to be at the right part of its orbit for the spacecraft to meet the ISS right after launch).
A launch site crosses the plane of the ISS orbit twice a day, but the phase angle at those crossings is often quite far off of optimal - you want the ISS to be a bit behind at launch so that by the time your spacecraft gets up to the right altitude/speed it's matched up. So you have to find the minimum phase angle at launch window for a targeted several weeks or so, then have the ISS make slight orbit changes to bring the phase angle closer to optimal. Once you're up and have as low of a phase-angle difference as possibly, you then have to spend a bunch of 90-ish-minute orbits getting your position in orbit matched up to the ISS.
At fastest, you can get down to the single-digit-hours range, but that takes a good selection of launch window, a bit of a burn in advance from the ISS to tweak its phasing, tracking systems that don't waste an orbit determining where exactly you are, and no scrubs.
This article is a good example of the process (but not the physics/geometry): https://www.nasaspaceflight.com/2018/07/progress-ms-09-super...
> The long sought attempt at launching a Russian Progress resupply vehicle to the International Space Station and having it rendezvous with the orbital complex after just two and a half orbits and just under 4 hours in space is being attempted.
...
> As with the other fast track rendezvous attempts, the reboost and phasing alignment of the Space Station with the Baikonur Cosmodrome is only available for this launch day. A scrub to today’s instantaneous launch of Progress MS-09 would have resulted in the craft having to fly a two-day, 36 orbit phasing profile to the ISS on a realigned launch date later this week.