I'm not jumping to defend because I was away and indisposed. Now I'm back and I have an answer (I hope) for you.
First I'd like to say 6,000 is not just a nice round number some YEC's like to use. And BTW, some YEC's are not dogmatic about the number 6,000 as some will say ~6,000 to 10,000. And for a more accurate depiction, it should be stated as ~6000 yrs old, but I am guiltly of dropping the approximate part.
Now, getting to where the ages add up. The geneologies in Genesis are what are counted up to give us a relative age of the earth. Archbishop James Ussher is the one credited with dating creation to 4004 BC. I got the following table from AiG. I don't think you guys should have a problem with this as it is a "Biblical" question of how creationists arrive at an ~6,000 year old earth. The full article is here:
Appendix B-The Forgotten Archbishop
"Answer: He (Ussher) took the chronologies in Genesis 5 and 11, together with some other Bible passages which we will look at. To simplify the calculations, we will tie the chronology to the fall of Jerusalem in 588 BC. The detailed calculations cover over 100 pages in the original document! From Genesis 5 we get the following:
First Genealogy-Genesis 5
Verse Event Age of the Earth
1:1-31 Creation = 0
5:3 Seth born when Adam, 130 = 130
5:6 Enos born when Seth, 105 = 235
5:9 Cainan born when Enos, 90 = 325
5:12 Mahalaleel born when Cainan, 70 = 395
5:15 Jared born when Mahalaleel, 65 = 460
5:18 Enoch born when Jared, 162 = 622
5:21 Methuselah born when Enoch, 65 = 687
5:25 Lamech born when Methuselah, 187 = 874
5:28 Noah born when Lamech, 182 = 1056
11:10 Shem born when Noah, 502 = 1558
7:6 Flood when Noah, 600 = 1656
From Genesis 11 we get:
11:10 Arphaxad born when Shem, 100 = 1658
11:12 Salah born when Arphad, 35 = 1693
11:14 Eber born when Salah, 30 = 1723
11:16 Peleg born when Eber, 34 = 1757
11:18 Reu born when Peleg, 30 = 1787
11:20 Serug born when Reu, 32 = 1819
11:22 Nahor born when Serug, 30 = 1849
11:24 Terah born when Nahor, 29 = 1878
11:32, 12:4 Abraham born when Terah, 130 = 2008
12:4 Abraham enters Canaan, 75 = 2083
In the Bible there are some large time periods given. These enable us to do the same calculations as Ussher, without going into all the intermediate details as he did.
Golden Arches of Time
Abraham left Haran until the Exodus exactly 430 years to the day.
-( Ge 12:10, Ex 12:40, Gal 3:17) = 2513
Exodus to start of Temple, 479 years
-( 1 Ki 6:1, in the 480th year or after 479 years) = 2992
Start of Temple to division of the Kingdom, 37
-( Solomon reigned 40 years, 1Ki 11:42, temple started in his 4th year) = 3029
Division of the Kingdom to final deportation about four years after Jerusalem fell,
390 whole years plus part of one year
(Eze 4:4-6) = 3421
Final deportation in 584 BC
Hence date creation = 584 + 3421-1 = 4004 BC
Now you have a rough idea of how Ussher did his calculations."
Edit- The graph is not coming out lined up properly. I would say go to the link to read the article on how creationists arrive at an ~6,000 yr old earth.