This is a fairly straightforward interpretation: Egeus says on multiple occasions that his daughter is his to give away in marriage, and her obedience is "due to him".
It would probably shock many modern readers to hear a father speak in this manner (i.e. do what I say, or I'll have you executed) but it should be taken into account that we are clearly meant to have some sympathy for Hermia, as well as to consider that Egeus has a reputation to uphold, and to have his daughter publicly reject his authority is, under the morals of the time, akin to treason and sabotage. Egeus would not only suffer the shame of having his authority ignored, but of everyone knowing that he had raised a daughter who refused to follow some of the most basic social guidelines, and that he had essentially failed to do his duty as a father.
We should also consider that Demetrius is clearly not a bad, insulting or dangerous choice; Egeus is acting in a way that he thinks is best for his daughter. Being loyal and obedient, in this context, doesn't just mean "do whatever I tell you to do", it means "trust that I can guide you to happiness and success". Egeus is probably feeling deeply insulted that his daughter, admittedly a young and naive girl, thinks she knows better than her noble and respected father when it comes to choosing men.
A Midsummer Night's Dream was written for Queen Elizabeth. Because of this, you have to take Egeus' comments with Elizabeth's perspective of women's rights when it comes to marriage. The play itself has a theme running through it discussing law vs love.
Now Hermia is the embodiment of love. Her father, Egeus, is the embodiment of law. Much of Midsummer Night's Dream has characters that represent concepts and ideas as well as being figures in the play.
Hermia is obsessed by Lysander. It is an irrational, imaginative fixation that has her father frustrated. (It does not follow traditional reason.)
Egeus is the epitome of law. Demetrius in his opinion is the better match and has the father's blessing. Because of the law, Egeus sees it as his right to control who his daughter marries. This clash ends up with him threatening to put her to death or in a convent.
Because of Egeus' "Law oriented" character fixation, Hermia should be, in his opinion, Loyal and obedient. But... she isn't. She is headstrong and chooses to go her own way. Egeus, being law, makes the general statement that government would like those that it controls to be loyal and obedient.
This interplay was Shakespeare's inside joke regarding Elizabeth's choice to be the "Virgin Queen" against the tradition of marrying and putting a man in power. Shakespeare is making the statement that it is impossible for the men in the court and religion to control a headstrong young woman, like Elizabeth. She, like Hermia, ends up getting her way. Egeus is overruled by Theseus (the supreme law of the land) in the end, just like Elizabeth through God retains her throne until her death.
Because he says so and because he is willing to have his daughter sentenced to death or a nunnery if she continues to disobey him. This was typical of the attitude toward women in Elizabethan and pre-Elizabethan Europe. It is seen in Romeo and Juliet to a devastating degree. Shakespeare contrasts this father/daughter dynamic with other male/female dynamics, all quite different, in Midsummer, one of the greatest relationship plays of all time.