Even if you are the only one on this network, and that you are of course a trusty node, mining is important for 2 things: confirming your transactions and creating new coins.
The transactions are included into blocks that are mined, and since you are the only one on your private network, you also got the reward that you successfully mined the block. An interesting thing you may have noticed is that Ethereum creates a new block even if there is no new transations. From this link, you can see the answer:
Those empty blocks are constantly adding new layers of difficulty to
ensure the practical immutability of past blocks.
Moreover, you can find more information here.