Why mine?

Money supply

It's a way of limiting the money supply in a decentralised system. You don't want anybody to easily create new money - no-one would trust a currency where anyone can make themselves a millionaire at whim.

Most of the conventional ways of limiting money supply require a central authority who manage the printing of banknotes and their destruction and a regulatory system that prevents banks declaring that they hold more money than they actually do.

Paying for transactions

An electronic payment system consumes processing resources to validate the transactions and to make the appropriate records available to anyone who needs them. There has to be some ways to pay for this.

In a decentralised system, both transaction fees and mining are ways in which this work gets paid for.

Actually, bitcoin is peer-to-peer transaction system, it means there is no central authority which can assure the validity of new transactions before adding them as a block to the blockchain. Here comes miners as rescue, these are the special nodes in the network which calculate a nonce (this nouce is a 32-bit number which when input with the hashing algorithm, produce a hash that is partially predictable like starting with five 0s) as proof-of-work and send it to other nodes and the block hash get validate and appended to the blockchain (the miner who calculated the nonce and hash correctly gets the right to add the new block of transactions to the blockchain), in return miner get new coins as reward, as in turn maintains the incorruptible blockchain.