When you stake your KSM through Karura Liquid Staking, you will receive a share representation in LKSM account balance, which entitles the owner to a likely increasing quantity of underlying assets. These staked KSMs are used to stake would earn Kusama nominator rewards, and incur punishment (e.g. KSM being slashed) in case of validator found to be misbehaving (e.g. not maintaining required uptime).
The balance of the two would increase or decrease the amount of the underlying asset of LKSM. This means earning staking reward is as simple as holding an LKSM token, while LKSM is cross-chain capable and can be used to participate in other network activities such as lending or as collaterals to mint kUSD.
Proxy staking is used before XCM-enabled fully autonomous staking is ready. LKSM Proxy account operated by the Karura Council can perform a limited number of actions: stake, unstake and rotate validators on behalf of the trustless Parachain Reserve account (and subaccounts). However it cannot perform transfer or other operations which help safeguard the LKSM protocol. Read more on Proxy Accounts here.