I only included the 2009-2018 stretch and only later rounds (ECF + Finals) in order to omit his poor offensive seasons so the sample looks really good for LeBron [Just kidding, I have no biases here and want all the data to be available for transparency purposes

Note that rORtg is offense relative to opponent DRtg. So any positive number means your team played better offense than an average team did against this opponent in the regular season. Around +5 is league best offensive level and +10 is historically great offense.
2009 ECQF vs Detroit: +5.0 rORtg
2009 ECSF vs Atlanta: +11.4 rORtg
2009 ECF vs. Magic: +8.7 rOtg
2010 ECQF vs. Bulls: +10.2 rORtg
2010 ECSF vs Celtics: -0.8 rORtg
2011 ECSF vs. Celtics: +8.8 rORtg
2011 ECF vs. Bulls: +2.9 rORtg
2011 Finals vs Mavericks: +2.9 rORtg
2012 ECQF vs Knicks: +11.9 rOrtg
2012 ECSF vs. Pacers: +3.5 rORtg
2012 ECF vs. Celtics: +10.7 rORtg
2012 Finals vs Thunder: +11.9 rORtg
2013 ECQF vs Bucks: +5.8 rORtg
2013 ECSF vs Bulls: +10.9 rORtg
2013 ECF vs Pacers: +12.1 rORtg
2013 Finals vs Spurs: +7.6 rORtg
2014 ECQF vs Bobcats: +9.8 rORtg
2014 ECSF vs Nets: +11.2 rORtg
2014 ECF vs Pacers: +18.7 rORtg
2014 Finals vs. Spurs: +2.4 rORtg
2015 ECQF vs Celtics: +7.1 rORtg
2015 ECSF vs Bulls: +8.0 rORtg
2015 ECF vs Hawks: +10.9 rORtg
2015 Finals vs Warriors: -1.8 rORtg
2016 ECQF vs Detroit: +14.9 rORtg
2016 ECSF vs Hawks: +21.5 rORtg
2016 ECF vs Raptors: +13.3 rORtg
2016 Finals vs Warriors: +5.3 rORtg
2017 ECQF vs Pacers: +11.0 rORtg
2017 ECSF vs Raptors: +12.5 rORtg
2017 ECF vs Boston: +20.0 rORtg
2017 Finals vs Warriors: +10.6 rORtg
2018 ECQF vs Pacers: -2.6 rORtg
2018 ECSF vs Raptors: +21.4 rORtg
2018 ECF vs Celtics: +2.5 rORtg
2018 Finals vs Warriors: +0.9 rORtg
2020 WCQF vs Blazers: +0.6 rORtg
2020 WCSF vs Rockets: +6.5 rORtg
2020 WCF vs Nuggets: +7.1 rORtg
2020 Finals vs Heat: +8.6 rORtg
There are several possible explanations:
a) Lebron, being the pseudo offensive anchor of these teams, is a wildly consistent as an offensive anchor. His individual excellence or effort on the offensive end largely contributed to these results.
b) The teams' offensive makeup was perfect. For instance these teams not having traditional big men to clog the paint relied more heavily on spacing. Fatigue late in the playoffs was non-existent as LeBron teams would routinely ramp up during a playoff run.
c) The teams were offensively slanted in essence sacrificing defense for better offense.
Honestly I think it's a combination of all three reasons.