Egeus is angry with his daughter Hermia because she is disobeying his will for her to marry Demetrius. Because Hermia is disobeying him, he is petitioning Theseus to sentence her with Athens' "ancient privilege," meaning the right to either kill her or send her off to a convent (I.i.42). However, it should be noted that while sending her to a convent is an option, Egeus is asking permission to "dispose of her" either through marriage to Demetrius or through death, rather than sending her to a convent (43-46).
It should also be noted that Shakespeare seems to be questioning Egeus's decision to have Hermia marry Demetrius in stead of Lysander. Egeus accuses Lysander of seducing his daughter and "betwitching" her through poems, "love-tokens," and singing insincere love songs beneath her window by the moonlight (28-32). In short, he is accusing Lysander of not truly loving his daughter but merely trying to seduce her. However, Shakespeare also points out that the sincerity of Demetrius's love is actually what Egeus should be suspicious of. Lysander argues that Demetrius is actually the insincere one because he has already seduced Helena and won her heart and "soul" to the point that she "dotes" upon Demetrius, "Devotedly dotes, dotes in idolatry" (110-111). Theseuss even confirms Lysander's accusations of Demetrius by saying that he has heard the same thing (113). We actually learn towards the end of the play that Demetrius not only seduced Helena and made vows of love, but he was actually engaged to her. Hence, Shakespeare is pointing out that Demetrius's character is questionable, showing us the questionability of Egeus's decision to want him to marry Hermia.
Aside from Demetrius's character being questionable, we also learn from Lysander in this scene that he is equal to Demetrius in social status and possibly even wealthier, as we see in his lines:
I am, my lord, as well derived as he,
As well possess'd; my love is more than his;
My fortunes every way as fairly rank'd,
If not with vantage, as Demetrius'. (101-104)
Hence, Shakespeare is showing us that there is no rationality behind Egeus's decision to want Demetrius to marry his daughter instead of Lysander and no rational reason to have her killed should she disobey. Egeus's anger and his decisions help portray Shakespeare's theme of questioning the rationality of man.
Egeus is angry because his daughter, Hermia does not want to marry Demetrius, the man her father arranged to marry. She refuses because she's in love with Lysander.