The term “Carry Forward” simply refers to content currently available to players in Modern Warfare II and the current version of Call of Duty: Warzone (such as weapons and cosmetic content, including Bundles, Operators, and other rewards and unlocks from the game), and how much of it will be accessible in Modern Warfare III.

Content previously described — weapons and cosmetic content, examples of which are shown in the above image — that has been previously unlocked or is currently available to players in MWII, will carry forward between MWII and MWIII.
Now for the caveats: