Because of the rules the gods have to follow while in the mortal realm, they are limited in the ways in which they can interact with it. The main rule is to not let loose their power and destroy big chunks of the realm or big numbers of people. In order to follow the rules but still have fun, the gods have to act indirectly.
The first solution is to talk to their mortal followers and tell them what to do. Unfortunately, mortals are both stupid and have this annoying thing called self-determination. And if you burn them all to a crisp or drown them in punishment, this does not make their corpses more obedient. Thus, when the task needs a more delicate handling, gods will take matters into their own hands. This is done thanks to possession.
Gods choose their mortal vessels with care. You will never see them—or at least admitting to their presence—in a not aesthetically pleasing body. They will also always prefer nobles, so that they can be given the respect they think they are due, even when not proclaiming their presence to everyone. Their favourite will always be the priests leading their own Church, such as the Church of the water god or the Church of the forging god.
To possess a mortal, gods take an immaterial form and superpose themselves with the mortal's body. They then shift their form so as to take hold of the body with their magic and inhabit it. Because of their power, a god's will always supersede that of a mortal, allowing them to move the body around without problems. Mortals stay perfectly aware of what's going on and what their body is doing, and they cannot escape into unconsciousness.
Mortals do not feel the emotions or thoughts of the god possessing them, but the gods are aware of everything going on in the mortal's mind at the time of possession. Unsurprisingly, it is very easy to anger the god by thinking something inappropriate. Even less surprising, few mortals survive their first possession. One need to have a very particular mindset in order to stay forever grateful for the god's presence, or excellent training in the control of their thoughts over long period of time.
Another danger is that the gods have to be constantly in control of their powers while inside their mortal vessels. This is very difficult to maintain over even an hour for most of them. If they slip a tiny bit, this never ends well for the mortals. While the gods certainly understand this basic fact, they tend to forget it every time they are roused up and attempt to use their full magical powers.
Beyond such drastic consequences, possession that lasts for too long leaves the mortals drained of energy afterwards, in some cases barely able to move for months. This is not because their body was damaged, but because it got used to working on a lot more energy and cannot cope anymore without it. Nevertheless, if they survive this first hurdle and recover, the mortals will then become very healthy and gain enhanced magic sensitivity and magic control. Most people let themselves be blinded by greed and think these rewards make it well worth the risks.
While not everyone can notice gods possessing mortal vessels, for people sensitive to magic, this does make a subtle glow similar to the Brand gods sometimes let on mortals they have touched. Gods can go through more or less effort to dim that glow, especially if they want to go undercover to do something twisted. Nevertheless, many gods do not really bother to do much beyond insuring that most commoners cannot see them and bother them.
For those of us able to notice their presence despite those efforts, the gods are everywhere and inescapable. This makes it even more obvious that our kingdoms and politics are nothing but a game for them and us their puppets. And even whatever illusions of control we have could be taken away from us at any moment...