So far, I've understood the meaning and derivation of centripetal acceleration as $v²/r.$ Centrifugal force is therefore basically mass times the centripetal acceleration. But how exactly is the connection made, so that we know the inertia of an object is pulling it outwards with that force?
Centripetal acceleration is the acceleration required to sustain circular motion, and it can be caused by any kind of force -- tension, normal contact, gravitational, electrostatic, etc. Centrifugal force is a fictitious force that appears to be present when you analyze the situation in the non-inertial reference frame of the object in circular motion: when you try to do calculations in reference frames that are accelerating, additional forces show up that are only due to the motion of the system.
Centrifugal force is the fictitious force that is necessary to explain the non-movement of an object in its own reference frame despite whatever force is (actually) causing its centripetal acceleration. If you're sitting in a centrifuge, you're aware of the normal contact force when you're sprawled on the edge of the chamber, but you need a fictitious force balancing it to explain why you're staying at the edge. You are trying to do physics in your own reference frame (after all, you are never moving relative to you!), and when you're accelerating that always leads to such complications.