For promotion, roll 1d10. If, after the listed modifiers to the roll, you roll equal to or greater than the number listed under Promotion, your character is promoted.
There's the answer, now let me walk you through it.
The easy part is to pin down what number we're trying to roll. Look at the wording: 6+, six plus. That's pretty clear that whatever you're rolling you want to roll a 6 or higher on it. Also note that a DM adds to your die roll, making it easier... so it would make no sense for a lawyer to get a +1 DM with if their EDU(cation) stat is 7+, if that only made it harder to get promoted? So we're obviously dealing with the promotion roll being the listed number or higher on whatever sided die we're using.
Note also that changing military branches imposes a -2 to your roll:
Changing Branches: Characters who are planning to change military branches on the following term must subtract 2 from their promotion die rolls.
(p.30)
It is decidedly more difficult to get promoted from O2-O3 in the Navy if you just made O2 in the Army; you're busy learning how this Navy thing works, vs the O2 who's been Navy from day one who already knows that crap. So no, if the promotion roll is listed as "6+" as in your example, the goal is absolutely not to roll 1-6. The goal is to roll 6-10 or 6-20.
As some noted, let's take a look at 2.0:
In 2.0, which 2.2 is heavily copypasted from, the rule was thus:
Rank
Promotion: A players roll versus Intelligence for promotion at the end of each career term in the armed forces. Graduates of a military academy subtract one from their die roll. Characters wishing to change arms of service in their next term add two to their die roll.
(p.20)
And for rolls themselves, it's all d10:
Rolling vs. (Against) an Attribute or Skill: To achieve success, the die roll must be less than or equal to the numerical falue of the attribute (unless otherwise speciifed). Die roll modifiers (if any) are to be made to the die roll before comparing the result with the attribute number.
(p.16)
Note that military academy grads subtract 1 from their die roll, making it easier to roll their INT or less; note that changing arms of service (which includes branches) add 2 to their die roll, making it harder. These are both the inverse of the 2.2 rules. So we know beyond the shadow of a doubt that the 2.2 rule is to roll the listed number or above.
But what freaking die do we roll!?!?!?!?!?!
Well this is where we look at 2.0 vs 2.2 again.
In 2.0 you can randomly determine attributes by rolling 2d6-2, rerolling any results of 0. This means a 5 is average; however, players who choose to use point-based allocation get 32 points, making the average just over 5.
Therefore the promotion roll in 2.0 would be, on average, a 50/50 chance, since we know 2.0 uses d10 (it uses no other die, save d6 for damage).
All 22 of the Civilian careers have a 6+ or 7+ for promotion rolls, and average up to 6.5454... pretty even split between 50/50 odds (6+ means a 6 or higher, inclusive) and 60/40 odds against promotion if we're doing a d10. That matches 2.0, more or less.
Of the 34 Military careers, all have 6+ for promotion rolls except 1: SEAL Officer, which has a 7. Again, that's 50/50 odds if we're using a d10, and again that matches 2.0 almost perfectly.
Add Civilian and Military careers together and average them out and you've got an average 6.232... so, a 6+ for the roll, on average. And again if we're using a d10, that's 50/50 odds on average, matching 2.0.
There's nothing listed in 2.2 about massively altering the promotion tables--and there's even a place for this, "Designer's Notes," p.276. The closest it comes to addressing rank is noting the decreasing skills-per-term from Traveller... and no, that's not coming very close at all.
Therefore, I can only conclude that no change from 2.0's rank progression is expected: The designers didn't mention it, and the numbers line up almost perfectly, with a little adjustment which should be expected from a version update. A massive change of the odds from 50/50 for an average character to 25/75 would certainly be worth noting!
So in the end, the answer is:
For promotion, roll 1d10. If, after the listed modifiers to the roll, you roll equal to or greater than the number listed under Promotion, your character is promoted.
Yes, it's silly that this wasn't simply stated. There's no reason to have to go through this much effort to figure it out--I actually incurred an expense on this answer! (To be fair, I'd been looking for an excuse to buy 2.2 PDFs for years.) This really should have been stated on page 30. I'm going to assume everyone involved just understood how it was meant to work--something that's easy to do with this kind of update.
On a sidebar, for civilian promotions, we are specifically advised that they do not necessarily mean promotion:
Civilian Promotions: Civilian careers have no ranks. Characters in these careers roll normally for promotions, and receive the additional skill when they roll a success, but no rank or change in rank is recorded. Players should assume that some form of professional advancement or recognition was achieved with the roll, but nothing so easily recorded as a military rank.
(p.30, emphasis mine)