As I remember it (tried it last year I think), the application needs to be in the foreground in order to do anything at all, because of Apple's (purposeful) limitations of doing things in the background.
So if you were hoping to be able to install this and sync stuff without effort and having to leave the app open all the time, Apple seems to be vehemently against anything like that, probably because they have their own solutions for this...
Edit: The GitHub repository actually goes through the Apple-induced problem:
> iOS is very much designed around foreground interactions. Therefore, background “daemon-style” applications don’t really exist under conventional means, so the behavior where KDE Connect iOS is unresponsive in the background is more or less intended. There are technically some special categories and "hacky" methods to try to get it to run in the background, but in general, there is no intended/by-design method of keeping a "daemon-style" app running forever in the background. For more information, see this post on the Apple Dev Forums
I remember this too. It happened to me about 90 seconds ago.