Lido's stETH is a token representing your share of staked ETH. When you hold stETH, you will earn daily staking rewards through a supply rebase, meaning that your total supply of stETH will grow. The supply rebase happens on a daily basis, around 12pm UTC.
What happens when you transfer your stETH to a different wallet?
stETH is fully transferable, and you can transfer your stETH between wallets, to friends, etc, as you would with any other token.
stETH staking rewards go to the address holding the stETH. This means that when you sell your stETH, or transfer your stETH to an address you do not control, you will no longer earn daily staking rewards.
You can transfer your stETH to any address you own to continue earning staking rewards. When doing this, your staking rewards will go to your new wallet.
You can track your staking rewards here: stake.lido.fi/rewards