How Can the Bitcoin Wallet Blockchain Size Be So Small?
Assuming you are talking about a full node,
It does indeed need to download the full blockchain, to verify it and build up a set of Unspent Transaction Outputs (UTXOs). But the thing is, once it's downloaded all those blocks, its not going to go back looking through them, it just uses the UTXO set to verify new transactions. All those old blocks are basically stored solely for sending to other nodes. So if we have a checkpoint at some block hash, we can say that all blocks before that are legitimate and stop storing them, just store the UTXOs, and then just keep the most recent blocks so we can still propagate them around the network. This greatly reduces the amount of disk space used.

Of course, there are also SPV nodes which just store block headers and rely on full nodes for specific transaction data, so they have blockchains in the megabytes rather than gigabytes.