Why did he begin with "I am Avraham's slave"?
Bava Kama 92b: This shows that one should tell his own disgrace before (others reveal it). 1
Hadar Zekenim, Riva: This explained why he needed to speak before he could eat - first he must do his mission.
Ohr Yechezkel Midos p.241: True greatness is not to aggrandize oneself in a Midah that he does not have. He did not want them to think that he is an overseer over Avraham's property.
Mishnas R. Aharon 1 p.139: This is not lest others reveal it. Rather, it is due to Midas ha'Musar. If not, he could have said "I am the officer authorized over all Avraham's property"!