"Finishing" is the process of taking an existing idea, demo, or production from DAW Stems (Ableton, Logic, Garageband, Pro Tools, anything!) and giving it that last 30-50% of detail oriented effort to create a song that is truly radio ready and can compete in today's music industry.
The process can involve, but is not limited to: re-arrangement of parts (mainly harmonically), or re-arrangement of the entire song (preparing a proper build to a chorus, as oppose to a hook simply going on and on etc,) 'beefing up' sounds, embellishing important parts in the arrangement, eliminating unnecessary sounds (to help the most important parts come through) instrumental and vocal editing, along with mixing and mastering.
This saves time in the studio, since the overall idea of the song is already there. Finishing is about embellishing, and complimenting the song to make the most true version of it - it can be, and to create the final vision of what that song was meant to be, rather than writing from the beginning and creating from scratch.
Many artists, from Passion Pit, to CHVRCHES, to even top 40 artists like Taylor Swift have done similar "Finishing" processes within their writing & production process.