Anything digital can be copied, indeed duplicated endlessly, while the Internet is full of bad people. It's clear that designing digital money to be exchanged anonymously over the public Internet was never going to be easy.
Bitcoin (and similar digital cryptocurrencies) have to solve a number of problems: security of the contents of your 'wallet', trust between buyer and seller, the integrity of the currency itself. Any robust solution is plainly going to be both complex and counterintuitive.
Bitcoin's key architectural innovation is the blockchain: a list of every transaction which has ever occurred. Transactions - as they occur - are broadcast across the peer-to-peer network, validated by each node, assembled (for a fee) by 'bitcoin miners' into a new block which is then rebroadcast (there's a kind of race to finish a new one), the new block being finally stacked by each full node onto its local copy of the ever-growing blockchain. The protocol provides mechanisms to ensure global consistency as divergences (forks) are quickly damped out.
Transactions are protected (signed) by private keys (permitting you to spend your own coins) and public keys - used to construct bitcoin addresses (like bank account numbers) to which payments are addressed, and also serving to validate signatures.
The mechanism is illustrated by this diagram from Satoshi Nakamoto's original short paper (PDF) which has taken me hours to properly understand.
|The bitcoin chain of ownership|
There are endless overviews of bitcoin which handwave about how it works. You will never understand bitcoin that way, because the reason it works is in the detail. Andreas M. Antonopoulos's book contains that detail and is accessible if you already know about public key cryptography, cryptographic hashing and digital signatures.
The book itself is focused on developers - plenty of code examples - and is weaker on the overall architecture and those essential usage models. However, if you read it alongside Satoshi Nakamoto's original paper and the Wikipedia article on bitcoin, then you will get there -- and be both amazed and impressed.