How does the network know my current balance?
The network doesn't know your current balance.

The bitcoin network doesn't use the concept of "balances".  Instead the Bitcoin ledger tracks assignments of value.  So every bitcoin that exists was initially claimed by a miner, and then there have been assignments of value ever since.  

Now a bitcoin client can show a "balance" for an address but that balance is simply the result of looking at all the transactions for an address and determining which payment assignments made to that address remain unspent.

An unspent transaction is referred to as an UTXO (unspent transaction output).

So a balance for an address is simply all the UTXOs for that address.   This can be determined from the blockchain.  This is a huge amount of transaction data but that's why the Bitcoin-Qt/bitcoind client uses LevelDB which is very efficient at retrieving from a dataset like the blockchain.