How does Electrum wallet list my transaction record?
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)