It's all about having the .lrc file, it gets read automatically (as long as it's named like the song file) by mpv player and many android players.
It's a simple text file containing the lyrics with timestamps. You can get them from deezer for example with certain downloaders. Or look for them on Soulseek.
Note that it's not really a Spotify streaming app, rather it uses your (free) Spotify account for playlists, search, recommendations, etc. and then goes and find the song you want to play on YouTube, and the lyrics from scraping websites I think. Pretty clever.