Here is Thomas Schreiner’s explanation (which aligns with Doug Moo’s, NICNT, 2nd ed., 411–13):
“The logic of verse 14 is as follows: If you are still under the era of law, then sin will rule over you. But since you are under the age of grace, sin will not have dominion over you. Paul declares that life under the Mosaic covenant spells subjection to sin. The history of Israel confirms a redemptive-historical reading; Israel was sent into exile during the Mosaic era because it did not obey the law. The prophets themselves acknowledged that Israel had not kept the law in the Mosaic age and needed a new p 327 covenant (e.g., Jer. 31:31–34; cf. Ezek. 36:26–27). That living under the law meant living under the power of sin is also apparent from other ὑπό (under) phrases in Paul (see Schreiner 1993b: 77–81). Those who are of works of the law are “under a curse” (Gal. 3:10). Galatians 3:21–25 suggests that those who are “under sin” are also those under the law, “under the pedagogue.” Those “under law” needed to be redeemed from their slavery by the death of Christ (Gal. 4:3–5). Those who are led by the Spirit “are not under the law” (Gal. 5:18), and thus those who are under the law are apparently not led by the Spirit but subjugated to the power of sin. First Corinthians 9:20 is no exception, since Paul makes plain that he is “not under law,” although he agrees to live under the law in some situations for the salvation of fellow Jews. So too, that Jesus was “born under law” (Gal. 4:4) is the exception that proves the rule. He had to be born under the law to redeem those in slavery.
In Rom. 6:14 Paul connects liberation from sin with liberation from the Mosaic era because these were inseparable in his theology. The Mosaic era was the age in which sin dominated. It does not follow that there was no grace in the Mosaic era, nor does it imply that all Israelites lived under the power of sin. Paul was well aware of the OT remnant that included prophets and godly people such as Abraham, Moses, Joseph, David, and Daniel. What we have here is a generalization. Paul describes what was generally the case in the OT economy, not what was invariably the case. What struck him was how signally Israel failed, not the small numbers who kept the law. The promises to the nation as a whole were unfulfilled, since Israel didn’t keep the law.”
Schreiner, BECNT, 2nd ed., 326–27.
I think Moo and Schreiner are on the right track, but their view should be further refined.
Understanding the “under the Law” to mean under the Mosaic covenant relieves the tension Schreiner is wrestling with regarding the existence of grace in the Mosaic era. It is the Mosaic covenant that is a law covenant. But there are people in living in the period of the Mosaic covenant who were saved by looking forward to the promises of the new covenant. They received grace even though the Mosaic covenant was not a grace covenant.
Similarly, the new covenant is a grace covenant, but the age of grace is not devoid of law because in the new covenant God’s law is written on the heart. Paul was careful, in 1 Corinthians 9, to acknowledge that he was under God’s law (universal creational law?) and the law of Christ (the law written on the heart in the new covenant, of which Christ is the mediator).

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