The grantee is the buyer, donee, or heir taking title. Today's grantee is tomorrow's grantor — every chain of title is a sequence of grantee-then-grantor links, with the same person showing up on both sides of consecutive entries.
For mineral curative, the grantee identification matters because subsequent transfers (lease, mineral deed, royalty assignment) only mean something if the grantee actually owned the interest at the time of conveyance. A grantee who was a minor, deceased, or fictional at the time named on the deed creates a gap that has to be cleared through guardianship records, probate, or quiet title.
County clerks index grantees the same way they index grantors — alphabetically by name within a date range. Modern systems also index by document number, but name search remains the primary tool for finding everything attributable to a person across time.