"Vous avez des carottes."
Translation:You have carrots.
What you hear is a liaison. In order to get that smooth euphonic sound in French, words ending in a silent consonant but followed by a word beginning in a vowel (or h meut) may carry that consonant sound onto the vowel word.
vous avez → vou z-avez
elles ont → elle z-ont
mes enfants → me z-enfants
You can find more information here.
Des carottes does not mean "the" carrots.
If it is translated at all you can put "some," though honestly nothing at all is usually best. It is the plural indefinite article. (un/une singular, des plural) This article doesn't exist in English. For example:
une carotte → a carrot
des carottes → carrots
This is not to be confused with the possessive (of + the) de + les → des : les jupes des filles → the girls' skirts.