Antinous insults Odysseus for numerous reasons:
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Antinous seems to have an air of self-importance, and believes he is far better than a beggar. For a beggar to even talk to him is considered an insult to him.
he pierced Antinous's chin with an arrow
Odysseus shot unsuspecting Antinous through the throat with an arrow from his mighty bow, as Antinous was eating an apple.
Antinous was wooing Odysseus' wife Penelope, and eating Odysseus' food, wine and livestock. Odysseus had also wished that Antinous would die before his wedding day, and Odysseus proved his own words. Further, Antinous had proved to be an ungracious man, abusing Odysseus whom he thought a beggar. Antinous was thus not worthy of mercy.
Antinous is wooing his wife Penelope. Also, Antinous is rude, obnoxious, and treats Odysseus poorly. When Odysseus is a beggar asking for food, Antinous not only gives him no food, but also throws a foot stool at him. Antinous also earlier previously encouraged Irus the beggar to fight Odysseus.