Blade Affinity is a mechanic where a line between Driver and Blade goes from nothing to a kind of thin blue belt of dust and it peaks at a solid golden line between the two.
Affinity is raised in game in general just by standing in close enough proximity to the Blade, and seems to increase further with auto-attacks and blade arts.
Now, how would this apply to Rex in Smash?
Rex is the active player, his Blade would be either in the background but always try to keep close like Pokemon Trainer or an intangible AI that follows close like Icies. As he lands attacks, more so with specials and smash attacks, affinity will strengthen. When the affinity is Maxed, Pyra will change to Mythra, which will give a stat boost and possibly strengthen specials similar to Cloud's Limit system. The third form of the Aegis blade, Pneuma, alongside Rex's Master Driver armor, then become the Final Smash. Either being the Level 3 special she's limited at in the main game or her Lv4 Special which is only accessible in NG+.