There are a couple things going on here.
Monstrous progression classes
Monstrous progression classes were introduced in Savage Species as a way to gradually gain the benefits of monster types that would normally only be available at high character levels (because they have lots of racial hit dice and/or high level adjustment). The Werebear template you're looking at is based on this type of progression class - it's a 9-level class that gradually gives you the benefits of being a Werebear.
Specifically, it's based on the Werebear template class and the Brown Bear animal class presented in this Savage Progressions article. The 9-level progression class you've linked in the question is just the 3-level Werebear template class and the 6-level Brown Bear class mushed together.
Using the combined class presented on the Realmshelp page, the way this works for level adjustment is that:
- The first column in the table (the one marked "CL") represents the total ECL of the character taking the class, including both racial HD and LA.
- Any level on which you gain a hit die (as indicated in the column marked "HD"), your level adjustment doesn't change.
- Any level on which you don't gain a hit die, you instead gain +1 LA. For some reason this isn't directly indicated anywhere on the table, you just have to infer it from the difference between your ECL and your HD.
Upon completing the entire class, the character would have 6 animal HD and LA+3, just like a vanilla Werebear that had been created ex nihilo, rather than using the progression class.
Level Adjustment (LA) vs. Challenge Rating (CR)
For some reason, the table on the Realmshelp page has included a column for CR. CR has nothing to do with character progression or LA; it's a number used by DMs to estimate encounter difficulty and calculate XP awards. The numbers included on the table are just the ones from the SRD/Monster Manual entry on lycanthropies:
By class level or base creature, modified according to the HD of the base animal: 1 HD or 2 HD, +2; 3 HD to 5 HD, +3; 6 HD to 10 HD, +4; 11 HD to 20 HD, +5; 21 or more HD, +6.
You don't use this number at all when progressing a PC or calculating your ECL. That said, to directly answer the question in your edit, the numbers in the CR column overlap (don't stack); for the last two levels of the class, an NPC Werebear's CR would be increased by a total of +4, not +23 (the sum of all the numbers in that column).