During the Dry Ice cleaning process, contaminants are removed from equipment by a 'blasting' process.
During this process, the dry ice can be thought of as essentially a gas which evapourates/disappears on contact with the item to be cleaned. People sometimes think the contaminant disappears also, but given the contaminant is a solid substance it can't disappear like the dry ice.
All cleaning processes involve the relocation of dirt. When a floor is mopped, the dirt moves from the floor to the mop and from the mop to the water in the bucket.
With dry ice cleaning, the contaminant moves from the equipment, where it is unwanted, to an area where it can be better dealt with.
If the contaminant is a dry substance, when it is dry ice cleaned it generally falls to the floor where it is swept away or vacuumed during normal maintenance.
If the contaminant is a wet substance like grease, you take a methodical approach similar to hosing down a driveway. You start at one end and guide the grease to the other end where it is sucked / mopped up.