As members of lower nobility, you know that peers of the realm are just like the gangs in prison. You make homage to the biggest one, and keep yourself safe under their wing, but it's best that you do it properly, of course.
As Paradox have explained in their latest Dev Diary, Homage represents the formal pledge of loyalty by a feudal lord to their Liege and provides Opinion, Renown, and Prestige to both parties, providing the occasion goes smoothly. This decision is also available to AI as well as players.
This decision costs some Prestige to initiate, is only available to Feudal and Clan vassals, and can only be undertaken once per Liege. If the ruler changes or dies, you can make a new pledge of allegiance to their replacement.
As with Petition Liege, this decision can be rejected outright and is not totally without risk even if they let you through the door. There is a great deal of pressure for such an important event to go well, and not all lords and ladies are made for public speaking. Things can go awry in a variety of ways, depending on the Petitioner’s skills and other factors, for example, if you've put in the hours to truly learn Norman, or merely practiced a few key lines the night before on the road to Windsor? Will your Shah empathize with your stammer, or imitate it in front of the entire Persian court? Will you forevermore be called ‘the Clumsy’ by your vindictive Sultan, simply because you lost your balance once, as can be seen in the example given to us by Paradox.
If something does go wrong, it is up to the Liege to decide if they will let it go, and giving your rival the opportunity to publicly mock your clumsiness might not be the best idea. If they do decide to embarrass you, you will lose some Prestige and Opinion with your Liege, instead of gaining them.
Regardless of the outcome, both your Dynasties will gain a small amount of Renown, and the rewards are increased by offering gifts in addition to your pledge, such as a Contract change in favor of your Liege.