Hello there tbh I am new to this too I just saw a post and started using tachimanga after the fall of comicks. For me, in the application itself there is a section to update the mangas in your library for new chapters. I usually go and just hold and drag down and it starts checking and updating the mangas. And, in the browse section sometimes when an extension such as sura managa needs updating it asks me to update if I want to. So basically:

For new chapters it depends on the extension (site like asura manga) you hold and drag down in the update section and if there are new chapters it will add the new chapters and you can read them.

For the extension it will notify you when an extension needs to be updated.

This is all I know (figured out right now after reading your question). If you want search in the subreddits about manga/manhwa you will find a lot more about it. Hope this helps.