Why do I need to report to the police when I visit Indonesia?

Some countries require that foreigners register their stay with the local police. Normally, if you're staying in a hotel then this is taken care of for you. If you are staying at a private residence then you must do the registration.

For example, from Entry Requirements - Indonesia:

Registering

If you stay in private accommodation in Indonesia (not a hotel) you must register your presence with the local police or you could face a fine of Rp 5 million [~ 375 USD]. If you stay in a hotel you will be registered automatically.

In Indonésia, when you stay more than 24 or 48 hours at someone's place, you have to register to some authority (not exactly the police, but I forgot the official name).
This is not only foreigners. The police just want to keep track of people (or at least, pretend they do)

You don't need to tell them anything. Just give your ID and the address you'll stay at. They will ask you plenty of question that you may find nosy, but that's mostly Indonesian chit-chat.

You don't have to pay anything, but Indonesian official being what they are, you possibly will have to. You know... just being polite with the elders :-/ .
Let your local friend deal with this part. It's easy to accidentally insult people by giving or not giving money to the right/wrong person.

Why should I report to the police?

They just want to record the data, so if something bad happen (earthquake, volcanic eruption, etc) they will know who is missing. This is also applied if police or authorities demand data for investigation (terrorism or illegal foreigner)

What do I need to tell them?

Who are you, what business you came there, how long you stay, etc. You can say you are doing legal stuff for your marriage for example.

Should I pay anything?

Depend. But offcially you dont need pay anything.