For information on how the Greek grammar supports the New World Translation, there are some related answers over on Biblical Hermeneutics to the question:
In addition to an Appendix article of the New World Translation of the Holy Scriptures—With References edition, the Study Edition of the New World Translation also supports John 1:1 saying:
the Word was a god: Or “the Word was divine [or, “a godlike one”].” This statement by John describes a quality or characteristic of “the Word” (Greek, ho loʹgos; see study note on the Word in this verse), that is, Jesus Christ. The Word’s preeminent position as the firstborn Son of God through whom God created all other things is a basis for describing him as “a god; a godlike one; divine; a divine being.” Many translators favor the rendering “the Word was God,” equating him with God Almighty. However, there are good reasons for saying that John did not mean that “the Word” was the same as Almighty God. First, the preceding clause and the following clause both clearly state that “the Word” was “with God.” Also, the Greek word the·osʹ occurs three times in verses 1 and 2. In the first and third occurrences, the·osʹ is preceded by the definite article in Greek; in the second occurrence, there is no article. Many scholars agree that the absence of the definite article before the second the·osʹ is significant. When the article is used in this context, the·osʹ refers to God Almighty. On the other hand, the absence of the article in this grammatical construction makes the·osʹ qualitative in meaning and describes a characteristic of “the Word.” Therefore, a number of Bible translations in English, French, and German render the text in a way similar to the New World Translation, conveying the idea that “the Word” was “a god; divine; a divine being; of divine kind; godlike.” Supporting this view, ancient translations of John’s Gospel into the Sahidic and the Bohairic dialects of the Coptic language, probably produced in the third and fourth centuries C.E., handle the first occurrence of the·osʹ at Joh 1:1 differently from the second occurrence. These renderings highlight a quality of “the Word,” that his nature was like that of God, but they do not equate him with his Father, the almighty God. In harmony with this verse, Col 2:9 describes Christ as having “all the fullness of the divine quality.” And according to 2Pe 1:4, even Christ’s joint heirs would “become sharers in divine nature.” Additionally, in the Septuagint translation, the Greek word the·osʹ is the usual equivalent of the Hebrew words rendered “God,” ʼel and ʼelo·himʹ, which are thought to convey the basic meaning “Mighty One; Strong One.” These Hebrew words are used with reference to the almighty God, other gods, and humans. (See study note on Joh 10:34.) Calling the Word “a god,” or “a mighty one,” would be in line with the prophecy at Isa 9:6, foretelling that the Messiah would be called “Mighty God” (not “Almighty God”) and that he would be the “Eternal Father” of all those privileged to live as his subjects. The zeal of his own Father, “Jehovah of armies,” would accomplish this.—Isa 9:7.
TL;DR: It states that the grammar of John 1:1c is in stark contrast to the surrounding verses which use a definite article ("the god"). The particular formulation used in John 1:1c is used in Greek to indicate a quality of the subject. It describes the nature of "the Word" and not the identity. It, therefore, emphasizes the mightiness that "the Word" possesses. This is further supported by the early Sahidic Coptic translation which uses an indefinite article in John 1:1c to form "a god," emphasizing the qualitative nature of the clause.
For an extremely thorough examination of the Greek grammar of John 1:1 that also examines bias in several common Bible translations (including the KJV, NIV, and NWT), read Truth in Translation: Accuracy and Bias in English Translations of the New Testament by Jason D. BeDuhn. Chapter 11 focuses on John 1:1, and comes to the conclusion that the New World Translation is the only translation he discusses that captures the qualitative distinction given to "the Word". He argues that the best translation of John 1:1c would be, "And the Word was divine." (The NWT puts "divine" as an alternate rendering of "a god" in the footnote for John 1:1c.) Professor Jason BeDuhn also refutes several common arguments that translators use for translating "the Word was God," including the misapplication of Colwell's Rule.
In Philip Harner's scholarly article on Greek grammar, "Qualitative Anarthrous Predicate Nouns: Mark 15:39 and John 1:1," published in Journal of Biblical Literature, Vol. 92, Philadelphia, 1973, p. 85, he suggests John 1:1 would be best translated as "the Word had the same nature as God." He criticizes the RSV and The Jerusalem Bible for translating "the Word was God" because "the English reader might not understand exactly what John was trying to express." Philip Harner establishes his grammatical observation in his conclusion (emphasis mine):
In interpreting clauses of this type it is important to recall that Greek writers also had other types of word-order available. If a writer simply wished to represent the subject as one of a class, he could use an anarthrous predicate noun after the verb. If he wished to emphasize that the predicate noun was definite, he could supply the article. The availability of these other types of word-order strengthens the view that in many instances we may look primarily for a qualitative emphasis in anarthrous predicate nouns that precede the verb.
A summary of reasons why "a god" is appropriate in John 1:1 can also be found on Wikipedia.