I guess this sentence means (エサを) あげているから (ペット/ 動物が 人に) 懐いている and this あげている means 与えている = "give" in English. The English translation is: (The pet/ animal) attaches to someone because (someone) gives (the pet/ animal) food. "give" has some translations in Japanese depending on the situations.

(someone Xが)(someone or animal Yに) (somethingを) あげて+いる+から、(Yは)(Xに)懐いて+いる。 懐いてる is abbreviation of 懐(なつ)いている . Dictionary form is なつく. The main verb of this sentence is 懐いてる. あげてる is another verb, but it just explains the reason for 懐いてる。 I'd say this sentence is incomplete because the verbs are missing WHO, WHAT, and TO WHOM. Ryoko-san's guess looks reasonable to me, though.