Study to shew thyself approved unto God, a workman that needeth not to be ashamed, rightly dividing the word of truth. - 2 Timothy 2:15 (KJV)
That's how the King James puts it. And no doubt, many of us have memorized (or at least, stored a paraphrased version of) this verse from the KJV. In particular, the part to "rightly divide the word of truth." Now I don't know about you, but for me, having spent my entire life in the Bible Belt, I've heard plenty a' Southern preacher proclaim this verse as a means of explaining dispensationalism. Namely, that the need to "rightly divide" the word, means to understand the difference in what was for then/them (Jews of the OT era), and what is for Believers today.
As Torah-observant believers, we already understand that Paul would not have been instructing Timothy to teach new believers that the Torah, or the Tanakh as a whole, was not intended for them. Or that they needed to carefully decipher which parts to keep, and which parts to cut out (as in the image above). Yet that message is clearly seen in some popular Christian teaching. Here's a quote from Joseph Prince on the matter:
It is crucial for every believer, when reading the Bible, to rightly divide the Word, and to clearly separate what belongs to the old covenant of law and what belongs to the new covenant of grace. When people quote Old Testament passages without appropriating the cross of Jesus in their interpretations, they make it seem as though the cross of Jesus Christ made no difference at all, leading to much misunderstanding and misinterpretation of the Bible. (from this article here)
Famed Christian theologian and dispensationalist, C. I. Scofield, even wrote a whole book on the matter, 'Rightly Dividing the Word of Truth (2 Tim. 2:15) Being Ten Outline Studies of the More Important Divisions of Scripture.' Scofield uses the verse as a springboard to discuss the different divisions and dispensations of God's grace, and how the Scriptures align within the framework of each division. (At least, according to how he interpreted Scripture, and his view on Dispensationalism).
But if that doesn't sound like something Paul, the Jewish Apostle and Pharisee would have in mind, well, don't fret; you're not alone. Here's a few quotes from other commentaries on the matter:
this same workman (specifically, Timothy but by application today all believers) was to be accurate in delivering the message of truth. The truth is the gospel. Paul showed concern that Timothy would present the gospel without perverting or distorting it. He was not to be turned aside by disputes about words or mere empty prattle. 
We see this further played out in other writings where Paul addresses his concerns that people are perverting the Gospel of Messiah Yeshua. In his address to the wizard Elymas in Acts 13, Paul accuses him of perverting the "straight ways" of God. What was Elymas doing? He was trying to convince the proconsul, to persuade them away from the faith. As another example, Paul wrote much of Galatians to address the perverse Gospel message that was spreading: namely, that ritualistic circumcision must predate salvation by faith. Galatians heavily addresses this, and in some ways aggressively (as Paul says in Galatians 5:12, that he wished those who were causing such trouble would castrate themselves!). In 2 Corinthians 4:1-6, Paul again says he does not teach in secret, nor in hidden things and meanings, nor does he make subversive use of Scripture (the NASB says "adulterating the Word of God").
So clearly it is well within Paul's teaching framework to be concerned with people not just teaching Scripture and spreading the Gospel of Yeshua, but that it was done correctly, with the proper doctrine, and in context. And clearly, Scripture can be used in an abusive way (that's a whole topic of its own). So Paul is warning Timothy to not utilize Scripture as a weapon with which to beat the Body of Believers. Or perhaps worse, to mislead them.
Another quote, this time from the IVP Background Commentary:
Jewish readers would have understood an exhortation to be diligent in representing “the word of truth” rightly as an exhortation to study God’s law, where his word was found (cf. Ps 119:43). Although Paul presupposes such investigation of Scripture (3:14–17), his emphasis here is on accurate representation of the gospel in contrast to the empty words of 2:14 and 16. 
Simply put, according to Keener above, Paul already believed his readers (though in particular, Timothy) would be studying the Scriptures. And again, in this way, he encourages and exhorts Timothy to handle the Gospel correctly, and not pervert it with quarrels about words (2:14) or godless chatter (2:16).
The Tree of Life Version thus translates this verse as:
"Make every effort to present yourself before God as tried and true, as an unashamed worker cutting a straight path with the word of truth." (TLV)
That makes the imagery of it a bit more vivid. And indeed, it is not outside the definition of the Greek ορθοτομεω (orthotomeo), a word occurring only once in Scripture, and usually translated "rightly dividing." Given the wording in the TLV, the footnotes make the connection to Isaiah 40:3, which Paul may or may not have had in mind when he wrote his epistle.
A voice cries out in the wilderness, “Prepare the way of ADONAI, Make straight in the desert a highway for our God." Isaiah 40:3 (TLV)
Albeit I personally find this less likely, as even the LXX does not use the word. We do find it, however, in places like Proverbs 3:6 in the LXX, where we read, "make her known in all your ways so that it may direct your ways." (Lexham English Septuagint) That is, that it may "make a straight path" for you. (Here, the context being Wisdom).
One more commentary before we summarize, because I think it wraps it all up nicely.
God bestows his approval on the one who exhibits truth, love, and godliness in daily living, and who correctly handles the word of truth. The false teachers were mishandling God’s words, using them for their own benefit. Timothy was commissioned to handle the words of God correctly. All preaching should present the truth clearly, cutting through erroneous ideas or inaccurate opinions.
Revisiting our earlier misquotation of the verse, we see pretty plainly that Paul's exhortation that Timothy "rightly divide the Word of Truth" is not about figuring out what's for the Jews, and what's for Christians. It's not about what was the Old Covenant dispensation vs what is the New Covenant dispensation. It's about teaching and preaching the Word of Truth, in truth. Not using it, as I mentioned above, as a weapon against others. Not perverting it to promote our own selfish schemes. And again, as Paul addresses in the verses immediately before and after verse 15, the one who rightly divides the word of truth is not the one who is "quarreling over words" (vs. 14) or engaging in "godless chatter" (vs. 16). Given the greater context, this is referring to teachers and preachers. Verses 14 and 16 could use their own dedicated posts, and perhaps I'll get to that eventually.
I hope and pray this study has blessed you. If you already held this understanding, well, either treat this as a witness, or pat yourself on the back. ;)
I myself struggled with this verse on account of how I have heard it taught through the years.
Shalom to you and yours.
 Thomas D. Lea and Hayne P. Griffin, 1, 2 Timothy, Titus, vol. 34, The New American Commentary (Nashville: Broadman & Holman Publishers, 1992), 215.
 Craig S. Keener, The IVP Bible Background Commentary: New Testament (Downers Grove, IL: InterVarsity Press, 1993), 2 Ti 2:15.
 Knute Larson, I & II Thessalonians, I & II Timothy, Titus, Philemon, vol. 9, Holman New Testament Commentary (Nashville, TN: Broadman & Holman Publishers, 2000), 286.