These individual transactions or records are not stored individually though (and this is where the "block" comes in). Each transaction gets sent to a node/computer but it doesn't get sent individually, it gets sent in a block. Each block works as a timestamped cell for all transactions that happened at that time. The ways these blocks are made are by very difficult mathematical equations called a cryptographic hash. It takes all the computers on earth about 10 minutes to solve this equation and so these blocks are made every 10 minutes. Because the math problem is so difficult, it is highly unlikely that any one person would solve two blocks in a row (and thus get to manipulate the system).
Every computer that is connected to this blockchain has every block and every transaction already in its memory. All of the blocks together are called the ledger. This is a lot of data (and it can take up to 24 hours to download all of the ledger) but after the first time, you won't have to do it again. This means that every node is constantly checking the validity of a transaction against the whole previous backlog of transactions. So if somebody is trying to tamper with what already happened (to show that they have more money than they actually do or maybe to show that they have credentials that they don't have), it would get flagged by every node as being a bad transaction. The individual transaction level security gets a bit more complicated. Also not every transaction needs to be stored as its own transaction. Many transactions can be compiled into one in the form of a Merkle tree.
This could go on for a while and the math is pretty complicated, but I think that's an ok introduction. Obviously, a lot of the value comes from a deep understanding of how the security works, but that would get more involved.
You can read more about it here.
You can also watch a video that explains Bitcoin and blockchain in some detail here.
Why it matters
Blockchain technology enables people to converge on a consensus of data even when people are dishonest and malicious. So the blockchain is basically a global distributed ledger where anything of value can be moved and stored securely. Trust is established by strength in numbers.
More than Bitcoin, there are many ways in which the blockchain can be used. Musicians can use a company like Mycelia to build contracts into their songs so that artists can sell directly to consumers without a label or tech company. This means artists get paid first before the labels (or even without them at all).
Reputation systems could be created so that instead of using rating agencies and credit rating services, trustless transactions will be easier.
There is the ability for blockchain to be used for intellectual property so that any shared intellectual property will be able to be found no matter where it spreads.
Companies like StubHub could use blockchain to verify that the tickets that are being sold are legitimate.
The Internet of Things will need blockchain technology to manage all of the daily transactions.
This is just the beginning and it will be interesting to see what actually comes of the blockchain.