That is the only way to get money out of thin air. And because blocks are hard to create, money is hard to create. All other money comes from people sending it to eachother.
For the signature question, we need to differentiate between two types of signatures. The first is the block signature, which is defined by having a bunch of leading zeroes. The second is an ECDSA signature, which is your more traditional cryptographic signature with a public key and a private key.
When coins are mined out of nowhere, they get assigned to a public key. I'm oversimplifying a little bit, but this is sufficient to understand. To spend coins from that public key, you need to sign a transaction saying something like "I send X coins to person A". This message is signed with the secret key that corresponds to the public key that owns the coins.
The really important part about the blockchain is that it prevents you from spending the same coins twice. So, if you only have 12.5 bitcoins, you can't sent 12.5 to Alice and then send the same 12.5 to Bob. You can create signed messages that claim both. Without a blockchain, it's impossible to tell which one is valid, because it's impossible to know which one came first.
But that's the magic of the blockchain. It's a history of transactions. If you try to spend the same money twice in the blockchain, we can very easily see that you've spend it twice. We also know which one is the valid original one, because it will be first in the blockchain. So we know to accept the transaction that's first in the blockchain, and to ignore the second one. (note, in Bitcoin, it's actually illegal to have two conflicting transactions in the same blockchain. The second transaction will never be included into a block at all).
And, as described early in the video, this history is very hard to re-write. If you decide one day later in Bitcoin that you want to actually undo your spend to Bob and instead spend that money back to yourself, you will have to outcompete all of the mining that happened over the past 24 hours to re-write the blockchain. Given that 'all of the mining' is hundreds of millions of dollars of hardware working non-stop, you will not be able to re-write the blockchain.