I know it sounds weird, but that's really it in a massively simplified nutshell.
Almost all modern cryptography is based on prime numbers. Bitcoin leverages this a bit. It's more complicated than just primes, but a pair of very large primes are the heart of it.
To give you a small idea of the difference between factoring the larger numbers, take a look at this quick and dirty example...
$ time factor 8958200452356091 8958200452356091: 8958200452356091 real 0m0.003s user 0m0.002s sys 0m0.000s $ time factor 258285118849492814941 258285118849492814941: 258285118849492814941 real 0m0.003s user 0m0.002s sys 0m0.000s $ time factor 2313769868514373261031453299297155631 2313769868514373261031453299297155631: 8958200452356091 258285118849492814941 real 0m7.087s user 0m7.084s sys 0m0.004s
The above is from a Linux computer that has the 'factor' program installed, which will factor rather large numbers fairly quickly. The number to look at in each of the 3 examples is the one after 'real'. So, I was able to test the primality of the first two numbers very quickly. (.003 seconds for each). However, attempting to factor the much larger 3rd number, which was generated by simply multiplying the smaller two numbers, you can see that the time increases by several orders of magnitude.
I was going to do a bigger example, For instance, 325928985563371356674384180771539 * 6663742992756402236816080367 = 2171906993684118463655790070573355681075427646504235890274813. However, I kicked that off over 2 hours ago, and it's still chewing on that 61 digit number. I'm actually kind of interested to see how long it will actually take.
The reason that primes are important is that there are tricks you can use to determine primality, but actually factoring a number is 'hard' from a mathematical standpoint. There are some shortcuts you can take, i.e., all prime numbers are odd, so you don't need to test those, which actually gets rid of half of the work. When numbers get really large, it takes a boatload of computational time to actually do the factoring. Almost all public-key cryptography is based on this hardness problem, among other things.
This is why I understand the concept of crypto currencies and crypto mining, and can see how you could monitize it to a certain degree. Really though, since there is no army backing the currency, it's 'value' is somewhat arbitrary. I wouldn't be at all surprised if BTC someday replaced tulips as a cautionary tale about the Madness of Crowds.
They love using non existent problems to try to control people.