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 perform 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.
Malbim: This showed his importance. Avraham has many slaves - I am his special slave, who oversees all his property.
Ha'amek Davar: He explained why he will not eat before doing his mission, for his master's mission is more important than himself.
Mishnas R. Aharon (Vol. 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"!
