Geez, if some professional chemist wants to jump in, please feel free.
Short, short answer:
It is all about purifying water more and more (the water of your vinyl cleaner) so that residue is not left behind when the water has evaporated.
Short answer:
Both ions and molecules are composed of atoms. Ions are entirely charged species which "cling tenaciously" (remain dissolved) in water. The most typical is in hard water, usually calcium and magnesium ions. But there are lots more. They remain dissolved until the water of your record-cleaning solution evaporates and leaves behind ions as residue salts--undesirable as noise during playback. The various resins, non-soluble polymers, are used to remove ions as a three-stage process. Thus triple de-ionized water.
Long answer:
Both molecules and ions are composed of atoms. Molecules are sharing of their electrons whereas ions do not share. Water molecules, being very polar although not charged, do exhibit a sort of charge-like character ("dipole moment") and therefore strongly attract ions (entirely charged species).
Ions come in the form of positively-charged "cations," where the electron(s) has been given away entirely. Or negatively-charged "anions," where the electron(s) has been entirely retained. Thus, ions strongly aggragate with other ions of opposite charge or with polar molecules, due to electrostatic force. The size of ions can be a single atom (some metal or mineral), several atoms (typically organic or carbon-based; there are others) or many, many atoms--bound in a chain-link like polymer-resin. The smaller ions can be hard to remove from water whereas resins are never dissolved.
(I hope I got this right.) Triple de-ionized water has been taken through a three stage process of removing ions--cation resin, anion resin, mixed resin.
Charged resins can be used to attract and hold the oppositely-charged ions in water when water passes over, around or through the resin. The resin can exist as positively charged (cation resin) or negatively charged (anion resin). The "mixed bed polishing resin" seems to be some sort of mixture of resins that is used as the last refining stage.
Heck, what do I know? I'm just a musician. Anyone, please feel free to correct my chemistry.