How did Satoshi manage to mine so many bitcoins being the first user of the system? I'm just learning about Bitcoin and if I understand its conception properly, mining bitcoins means validation of transactions by computing a hash which should satisfy some preset condition. In other words, mining == servicing network and getting money for it. 

If Satoshi was the first user, how did he earn his bitcoins if there were no transactions?

I understand, that in the very beginning some geeks were involved in the network, so maybe 5 or 10 persons' (first testers) transactions and their validation (mining) generated a lot of bitcoins, because of the low difficulty and high reward. But how did the very first coins get rewarded?
Blocks can include transactions, but it is not necessary for transactions to occur in order to create blocks. The only transaction that is required in a block is the coinbase transaction which is the transaction that creates new bitcoins and collects the transaction fees. This transaction is created by each miner individually for their block attempts (as it sends money to their own address which is different for each miner), so it'll surely be available. In fact, the first transaction other than a coinbase transaction occurred in block 170.
So, since transactions are not a requirement, block creation is limited differently, it happens by means of the difficulty. Miners create block candidates continuously, and then check each whether it fulfills the difficulty requirement. Whenever they find a block that surpasses the required difficulty, they have successfully mined a block. As the difficulty resets regularly to adapt to changes in the hashrate of the mining network, the blocks work out to be discovered approximately every ten minutes.
In the very beginning of Bitcoin, mining difficulty was very low as there were few people running their home computers to mine Bitcoin. Satoshi Nakamoto was the first one to run a computer for mining, and thus was able to mine a large portion of the early blocks.