Text: αὕτη δέ ἐστιν ἡ αἰώνιος ζωὴ ἵνα γινώσκωσιν σὲ τὸν μόνον ἀληθινὸν θεὸν καὶ ὃν ἀπέστειλας Ἰησοῦν Χριστόν. (NA28)
γινώσκωσιν is Present Subjunctive Active 3rd person plural. KJV renders the verb (with ἵνα and σὲ) as "that they might know thee . . . "
The word "they" comes from the fact that the verb is third-person plural, whereas "might know" is because the verb is subjunctive. However, the 1984 NWT translated this construction, "their taking in knowledge of you" and 2013 has "their coming to know you . . . "
I sometimes wonder why the NWT does not explicitly render the subjunctive like KJV, Byington (I think) and others. On the other hand, some just translate "that they know you" without making the mood explicit (see Mounce and NET Bible).
I've also had the question about why we translate a third-person plural verb with a possessive pronoun "their" instead of employing "they." But I could be missing something linguistically or syntactically with the construction.