@CC1084 the UI is slightly confusing here. This is improved in the Manuscripts 1.2 beta version we're releasing hopefully later tonight, but basically choosing styles with the tickmark in that citation style choice screen marks them as favourites. You then need to go and choose the style from the dropdown, and it should then stick. Alternatively, double clicking will both set a favourite and assign it to the document (confusing, I know, this is better in the next version!)
If you don't see the bibliography refreshing, you can also go to Format > Refresh Bibliography (it might be Refresh Citations and Bibliography, I don't 100% remember?). If it still doesn't stick, please check that the style you chose indeed includes what you would expect it to include (e.g. numbered citations). If the style is indeed correct and it doesn't stick, could you try reopening the document (this causes the equivalent of Format > Refresh Bibliography).
CEO & Co-founder of Manuscripts.app Ltd