There probably is no "saying"
The whole phrase "meaning of the saying" is not in the Greek manuscripts (at least not in Nestle-Aland nor Byzantine Majority), and thus is added by the NIV translators. No other major translations add those words, and they appear to confuse the issue in doing so (however, the discussion below will point to why they probably did).1
There is a slight variation of significance here between the two text types. The passage reads in the majority text (extra comments and significant variations with the Nestle-Aland text noted by bracket numbers):
Ταῦτα δέ ἀδελφοί μετεσχημάτισα εἰς ἐμαυτὸν καὶ Ἀπολλὼ διʼ ὑμᾶς
These things then, brethren, I have transformed to myself and Apollos, for the sake of you,
ἵνα ἐν ἡμῖν μάθητε τὸ  μὴ ὑπὲρ ὃ γέγραπται φρονεῖν
in order that in us you may learn the not above what is written to think
ἵνα μὴ εἷς ὑπὲρ τοῦ ἑνὸς φυσιοῦσθε κατὰ τοῦ ἑτέρου.
in order that not one above the one may be puffed up against the other
 μετεσχημάτισα (meteschematisa): This word here "is unique,"2 meaning it is a figurative use of the term that is unique to that use in Scripture, and rare in other literature. The word literally means "to change" or "transform," including "to feign to be what one is not."3 Hence the various translations of "in a figure transferred" (KJV), "figuratively transferred" (NKJV), "figuratively applied" (NASB), or simply "applied" (NIV/ESV).
 N-A spelling variation Ἀπολλῶν. Spelling variations in names are common in Biblical texts, but only sometimes significant (i.e. when the change may really point to another person).
 Here is where the missing phrase that was added by NIV would roughly be if it existed. More discussion about this point in .
 The ὃ (ho; "what") is a singular relative pronoun; N-A has a plural ἃ (ha; "what things"). Grammatically, the majority text is correct, because the word is the subject of the verb γέγραπται (gegraptai; "has been written") which is singular in form, and in Greek the number of the subject matches that of the verb (normally; there are exceptions for collective words). Here it is unlikely the variation changes meaning, because the singular would be looking at what was written as a whole, the plural at individual aspects of what was written within the whole.
 γέγραπται (gegraptai) is a perfect passive participle. Normally perfect passives are translated with the helper verbs in English of "has/have been," and thus "has been written." However, with this particular verb, it is common (and perfectly correct) to translate it as "is written." This is because the perfect tense carries the idea of a resulting state due to the action what was in the past performed, and in the case of something being written, then it remains written (unless of course all copies of it are destroyed). In a way, this word itself hints at the continued preservation of the text (though in and of itself, that would be pushing the meaning too far). What is important to note with this word is that "In the sphere of revelation, the written records hold this authoritative position, and gégraptai always implies an appeal to the indisputable and legal authority of the passage quoted [me: or in this case, referred to]."4 Normally it is applied to OT quotations in the introductory formula "it is written" with the OT quote following.
 This is the most significant variation, as the word φρονεῖν (fronein; "to have an opinion," or "to consider," or "to think")5 is missing in the N-A text. So it would read from N-A translated "you may learn the not above what is written," (i.e. "you may learn not beyond what is written"). This variation is what compelled the NIV translators (who are generally following N-A text) to attempt the resolution they did, as the τὸ ("the") is by itself, implying what follows is a "definite" something being referred to. So they are basing their translation "meaning of the saying" off of the presence of the definite article here that is making the following phrase definite. Thus the article is acting as a "substantiver" of the clause to make it into a direct object for the verb μάθητε (mathete; "you may learn").6 The construction is used before a "statement, quotation, or clause."7 It may be the NIV translators felt this should be treated as a quotation,8 and hence their translation. Another oddity in this variant is that the μη ("not") is used with non-indicative verbs, but γέγραπται (the only verb in association with it in the N-A text) is in the indicative mood. This leaves the verb that the μη is supposed to be modifying as elliptical (i.e. not present and needing to be supplied).9 In that translation the verbal idea of "go" is supplied by the translators.
In the majority text reading that has the trailing φρονεῖν, it is still functioning as a "substantiver" of the clause, but the clause is an infinitive phrase with this as a modifying clause because the φρονεῖν is in the infinitive form, and the neuter τὸ is regularly used in conjunction with the infinitive, especially when the infinitive is the direct object of a verb as here.10 The intervening μὴ ὑπὲρ ὃ γέγραπται joins that clause into the whole infinitive phrase (that is, the leading τὸ and the trailing infinitive φρονεῖν define the whole direct object). So they are not simply to "learn to think" but to "learn to not think above what is written." Note that this variant supplies the verb that μή was oddly missing in the other variant, as μή would be used with the infinitive form of verbs.
Conclusion: My particular bias for the majority text aside (disclaimer), simply working through the meaning of what it is saying leads me to favor the majority text reading as correct, as the phrase makes more sense to me in context with that reading. But I leave each of you to decide that for yourself.
The fact that γέγραπται always refers to authoritative Scripture otherwise leads to a strong conclusion that such is intended here. For this reason, some believe it cannot refer to the preceding part of the letter.11
I personally do not have (1) a bias to exclude the possibility that Paul is simply using the term γέγραπται in a non-technical sense here, or (2) could in fact still be using it as a technical term, but relating such authority to what he had just written (especially in light of his statement in 4:1-5, but v.1 particularly, where he is making a statement of his authority).
That being so (i.e. not having the bias to reject it referring to previous context or that context being regarded as scripture), I believe the first part of v.6 in reference to the figurative transfer he has made using himself and Apollos (which refers back to 1 Cor 3:1-4:5) is clearly in view for the point of vv.6-7 here. That is, do not think of any other believer (yourselves included Paul is saying) above what any believer is,
- "ministers" (3:5) relying on God for effects of that ministry (3:7)
- "fellow workers" (3:9) by God's grace (3:10) to build up believers (3:10-15)
- part of "the temple of God" in whom "the Spirit of God dwells" (3:16), that must be kept holy (3:17)
- not to be self deceived and a fool, but wise (3:18-20)
- not proud, for there is an equality among believers, and all are under God (3:21-22)
- respecting those serving Christ and being stewards of His word faithfully (4:1-5)
Such were Paul and Apollos, and so Paul has transformed himself and Apollos into this illustration of what each believer ought to be and what each ought to think of other believers.
For me, the answer of "what is written" is Paul's revelation he just made to Corinth about man's role in God's work.
1 This is a case, in my view, of why the NIV is not one of the better translations. Just a little too dynamic equivalent (i.e. interpretive) in its attempts to translate, and things like this creep in that to me just confuse the issues more. NOTE: don't read into that more than it says. I am not wholly against a certain level of dynamic equivalence in a good translation, I just think NIV goes too far, and the more literal translations keep that balanced.
2 William Arndt, Frederick W. Danker, and Walter Bauer. A Greek-English Lexicon of the New Testament and Other Early Christian Literature (Chicago: University of Chicago Press, 2000), s.v. μετασχηματίζω, 3. Hereafter referred to as BDAG.
3 Ibid., entries 1. and 2.
4 Spiros Zodhiates, The Complete Word Study Dictionary: New Testament (Chattanooga, TN: AMG Publishers, 2000), 1125. γράφω.
5 BDAG, s.v. φρονέω, 1-3.
6 Daniel B. Wallace, Greek Grammar Beyond the Basics - Exegetical Syntax of the New Testament (Zondervan Publishing House and Galaxie Software, 1999), 231.
7 Ibid., 237-238.
8 BDAG, s.v. γράφω, 2.b, refers to 1 Cor 4:6 (the N-A form) as "a traditional formulation," but without any reference evidence prior to the composition of Corinthians, so the "traditional formulation," if such does exist, may be based off this version of this Scripture. Other lexicons note this as an idiom, but none give any proof (as I would have expected BDAG to).
9 I found this oddity noted of the N-A reading in both John Peter Lange, Philip Schaff, Christian Friedrich Kling, and Daniel W. Poor, A Commentary on the Holy Scriptures: 1 Corinthians (Bellingham, WA: Logos Bible Software, 2008), 93; and A.T. Robertson, Word Pictures in the New Testament (Nashville, TN: Broadman Press, 1933), 1 Corinthians 4:6. As to μή being used with the non-indicative mood, see BDAG, s.v. μή.
10 Wallace, 602 n.40.
11 Lange, 93. This commentary believes if the previous part of the letter was meant that
it would have been προέγραψα, I have before written (comp. Eph.
3:3). According to Paul’s usage, the formula: “it is written,” refers
to the Holy Scriptures, especially to the Old Testament: since we find
no allusion to any New Testament, or to any life of Christ in any of
Paul’s writings, ... Undoubtedly Paul here has in mind, not individual
expressions of Holy Writ, but its collective tenor, which all points
to this truth: that all honor belongs to God; and that all
self-boasting, all cleaving to men, and priding oneself in men, must
be given up. This doctrine we find summed up in apophthegms like Jer.
9:23, to which reference has already been made. (93)