Based on the adult manga by Konomi Shikishiro.

How do anime composers write their music?

The most important thing would be good guidance of desired emotion and themes from the director.
The composer could get started before even seeing any of the film.
Eventually of course they will need to see it for themselves, so any software that can play the video file, within the timeline or not.
Compared to dialogue, foley and sfx, and some sound design, the music isn't as critical to be in perfect time with the picture (obviously this depends on the style of animation).
Thankfully there is an affect called Synchronicity, which is where our brain automatically "syncs" and connects sounds with movements in images, being exact isn't necessary.
Besides, depending when yo have come on board the project, there could, probably will be any number of re-edits made.
As you get more of the film, better renders etc, you'll want any software that can sync the audio timeline to video playback - protools, ableton, logic, cubase, possibly reaper and fl studio, there will be a long list.
Then you can start refining musical phrases and changes.
There could likely be a mixing stage afterwards where someone else may remix and even re-edit some of what you have done. Long story short, the main thing you need apart from direction and creativity are instruments/recorder/software that will let you get ideas down quickly and have access to the sound palette you are used to working with.
For music - this is definitely Ableton Live for me.