Jerome's answer was spot on! I'd like to give your question a shot, too... just in case I am able to add anything helpful.
People convert the letters into numbers, or rather intervals, because although the letters change for every key, the intervals of chords and scales do not. For example:
Two notes are cannot be a chord because they are only an interval, so the fundamental foundation of a chord must be three notes. Also chords can only be built by stacking thirds. This means that the most basic chord possible is a triad, meaning: 1-3-5 (Root-third-fifth).
Now the root can never change, but the 3rd and 5th can. So that gives us these variations.
Augmented triads are kind of rare; but if you wanna hear a good example, check the intro to "Stardust" by Nat King Cole.
Diminished chords are a lot more common; they often act as transitions between diatonic chords. Check out the Jazz standard "Bewitched" as an example.
Now, about the 7... The seven is just following the formula used to find triads, stacking 3rds.
Most common 7th chords:
Maj7 chord: 1-3-5-7
Min7 chord: 1-b2-5-b7
(Dom)7 chord: 1-3-5-b7
Notice that those 7th chords are built from the Maj and Min triads. Other 7th chords can be build from Dim and Aug triads.
So now about the 9, 11, 13... you may have noticed that while stacking thirds, we skipped over the 2, 4 and 6. A quick recap:
CDEFGABC = 12345678
Well 8 is the same as 1 again, C and C. If we stack thirds beyond the point of 1-3-5-7, well moving a third over 7 will pass over 8 and lands at 9, which is the same as two... check it out, CDEFGABCD = 123456789. C is 1 also 8. D is 2 and also 9. A third above the 2/9 is 4/11. A third above 4/11 is 6/13.
9, 11 and 13 are considered extensions or scale tones. as where 1-3-5-7 are considered chord tones. Anything in between a chord tone and a scale tone are passing tones.
Hope that helps.
P.S. A quick tip, try adding a 9, just as an example, to any chord you play. It's never a "wrong" note, unless you are playing with someone who knows to altered a functioning Dom chord.
All the best, dudes! Check out my youtube channel if you want to see some videos with weird chord voicings (and normal ones too) and other lessons :-).
I make albums and lesson videos: