Too Long; Didn't Read
The aim of the Ethernaut game is to reduce the balance of the contract to 0.001 ether. To take ownership we need to use the 'contribute' function to reach a balance higher than 1000 ethers. The owner's address is different from than the player's address. To send the transaction you need to confirm your operation in MetaMask and pay some small gas. Once the transaction will be executed you should see confirmation as above. So we can verify that we contributed to the contract and claim all funds.