Plagiarised from a now long gone Bitcoin forum post (I can't find the author's name, sorry.)
The number comes from the combinations of the initial block reward (50 coins), the target blocks per hour (6) and the halving period (4 years). There will be 10.5 million created in the first four year period, and then the reward will be halved while all other metrics remain the same. As the reward continues to half again with each four year period, the total number of coins issued will trend toward the mathmatical limit (as in a logrithmic) of 21 million. The numbers that define the outcomes are the initial reward, the target block interval, and the halving term. All of these were design decisions that resulted in the outcome of 21 million, not the other way around. The interval could have been 9 minutes, or 12, or anything; same with the halving period or the initial reward. The interval needs to occur quick enough to verify transactions within a rational amount of time, and be long enough for a huge future bitcoin network to propagate transactions and blocks without significant latency issues; but why 10 minutes and not 6 or 15? Mostly it was an arbitrary design decision, and 6 or 15 or 525 seconds or anything else would have worked.
And why 10000000 satoshis make up 1 bitcoin?
Decimalisation. 0.00000001 * 10000000 = 1.
