How does Electrum wallet list my transaction record? I'm a beginner of Bitcoin and blockchain. I use Electrum as my BitCoin wallet, and install it on both home PC and office PC.

I can see all transaction records on Electrum wallet of both home PC and office PC.

You know a new receiving address will be generated when I require a new  receiving bitcoin every time. I think that the new receiving address is generated  randomly.

I think that Electrum wallet will not record all receiving address I used.

so I'm very stranger that how Electrum wallet find my transaction records from  blockchain quickly, could you tell me?

A bad way is to calculate all private key based seed, and calculate all public key and my receiving bitcoin address based private key, then find all transaction based all my receiving bitcoin address from blockchain. it's a huge work!
Electrum is backed by a number of electrum-servers (electrumx is a more recent implementation). These servers maintain the full utxo set, and also serve information such as block headers to electrum clients.

Since both your electrum installations are using the same seed, they will always generate the same addresses.

When you start electrum, it will contact an electrum server to retrieve the latest blockchain information. It will then utilize a simplified payment verification (SPV) system to query the node for transactions relating to your addresses. As it realizes addresses have been used, it will generate more addresses using the seed words.

The process of how addresses are created from a common root is described in BIP32. The SPV protocol is described in the whitepaper, and also by MultiBit (another wallet that utilized SPV systems)