Shorena's this famous thread will help any one how to sign a message. Still, I like to give a try for explaining the purpose of signing a message as OP mainly concerned about it.
what is the purpose of making signed message.
Purpose of signing a message =>
proving ownership of a bitcoin address.There are lot of different occasions, we might need to prove ownership of a bitcoin address. One good example must be staking address. Refer
here, if already not aware of this.
Another example :
You have some funds in your bitcoin address and you want to buy a domain in this forum's auction board.
By signing a message, you can prove yourself that you are having some funds and you are participating in that auction not for just time pass. (in real life auctions, we may need to show bank's demand drafts as proof of funds).
What is the other way to prove ownership of a bitcoin address : by sending a small amount from that particular bitcoin address.
But signing a message has some unique benefits over 'other way of proving ownership' :
it has readability and it is completely free of cost.