In this example you're viewing the balance of your primary account (with index `[0]`). `Balance` is your total balance. The `unlocked balance` is the amount currently available to spend. Newly received transactions require 10 confirmations on the blockchain before being unlocked.
`monero-wallet-cli` is the wallet software shipped in the Monero archives. It is a console program, and manages an account. While a bitcoin wallet manages both an account and the blockchain, Monero separates these: `monerod` handles the blockchain, and `monero-wallet-cli` handles the account.
Pass in the transaction ID you want the key for. Remember that a payment might have been split in more than one transaction, so you may need several keys. You can then send that key, or these keys, to whoever you want to provide proof of your transaction, along with the transaction id and the address you sent to. Note that this third party, if knowing your own address, will be able to see how much change was returned to you as well.
Payments made to an integrated address generated from your account will go to your account, with that payment ID attached, so you can tell payments apart.
Replace `TXID`, `TXKEY` and `ADDRESS` with the transaction ID, per-transaction key, and destination address which were supplied to you, respectively. `monero-wallet-cli` will check that transaction and let you know how much monero this transaction paid to the given address.
Since Monero is anonymous, you won't see the origin address the funds you receive came from. If you want to know, for instance to credit a particular customer, you'll have to tell the sender to use a payment ID, which is an arbitrary optional tag which gets attached to a transaction. It's not possible to use standalone payment addresses, but you can generate an address that already includes a random payment ID (integrated addresss) using `integrated_address`: