One of the defining elements of Planescape is that there are powerful, influential organizations that try and influence how all people on the planes think — the Factions.
PC's are encouraged to be part of these factions, or to found sects of their own. This reinforces the Planescape idea of swaying hearts and minds to believe as the party members believe, necessitating an organization. So organizations are much more important in Planescape than they are in many games — influence over NPC's is part of how the party advances.
Patronage codifies a relationship between the PC's and the groups they help out.
Any group of NPC's working toward a common goal might be considered a "patron." This goal might be something like, "Save the town of Bildab from the Necromancer," and the NPC's working toward that goal might be "the people of Bildab." In Planescape, these are often factional goals, like " Awaken a Dead God" for the Signers, or "Take over Sigil" for the Fated, with the NPC's within that faction, and leading it, being the NPC's working toward that goal.
A patron is defined by a goal. A peaceful village with no threat doesn't have much of need for adventurers, and so they can't serve as a patron.
PC's should have goals that align with the goals their patrons, but this relationship can work in any direction. A PC whose goals are different from their patron's goals might want to seek a new patron, and a patron might consider aligning their goals more closely with a PC's goals if they really want to recruit that PC.
Patrons have plans to accomplish their goals, and those plans involve, at some level, adventurers — they involve dangerous missions in risky territory, frightening beasts, political subterfuge, or other specialized and dangerous missions. A goal like "Awaken a Dead God" might require a few intermediate steps, unusual artifacts, fights against opponents (such as the Athar), and other tasks that cannot be left up to some middling functionary.
These tasks are quests that are available to the PC's. As the party completes these quests, the patron gets closer to accomplishing their goals, and may offer new quests that help in securing them. PC's can choose to not go on certain quests, or to take quests from competing patrons as well.
By accomplishing these quests, the party works with the patron towards the patron's goals. If a party doesn't want to see those goals realized, they may want to consider taking different quests — just because a party member is a member of the Sign of One doesn't mean they want to accomplish the goal of "Awaken a Dead God!" A patron may have many, or even competing goals (an Anarchist, for instance, could work to both raise the profile of the Anarchists, and to destroy it), so they may have many different sorts of quests.
Rewards of Patronage
Completing a quest for a patron grants you rewards in the form of XP and treasure. You may also find such things while on a quest, but a significant portion of your reward comes in the form of payment from your patron. Your patron might also offer exclusive access to certain items, character options, or abilities. Additionally, as your patron's goals align with your own, you get to see your hopes and dreams enacted in the campaign world.
The more you do for your patron, the higher and more powerful your rewards, of course.