Just to expand a bit on the progression: it works because iii and vi sound a lot like I. E.g. in they key of C I (C E G), iii (E G B), vi (A C E). iii can be s

Just to expand a bit on the progression: it works because iii and vi sound a lot like I. E.g. in they key of C I (C E G), iii (E G B), vi (A C E). iii can be seen as Imaj7 without the root (C) and vi as Iadd6 without the 5th (G). So basically this is I-V-I-I with some decoration, being V-I the quintessential movement in western classical music.