Quai's approach to scaling is do divide the network into multiple chains (shards), each processing its own transactions in parallel to one another. Through the process of merge-mining, miners contribute their hash power to at least 3 chains at a time, greatly increasing the network security and reducing energy usage. New shards can be dynamically added, providing infinite scalability.
Quai's shards divided in zones, regions and prime (source)
There are 9 zones, each subordinate to one of 3 regions: Cyprus, Paxos and Hydra. When creating a Quai wallet in Koala, your address is assigned to one of the 9 zones. This assignment is permanent and cannot be changed. A future update will allow optionally choosing the zone on wallet creation.
Though zones are isolated from one another, it's possible to send Quai between then. When sending to a different zone in the same region, the transaction must travel up to the Region shard, then back down to the new zone. When sending to a zone in a different region, the transaction must travel up to the Prime shard, then back down the hierarchy. This makes such transactions take a bit longer to confirm, but atomicity is guaranteed. It's not possible to "lose" funds in the process.
Quai's network is EVM compatible. This means developers on Ethereum should feel right at home to build on Quai. Those used to transacting on Ethereum should also be familiar with the concept of smart contracts and gas fees.
All of this technology is conveniently abstracted away from everyday users. They do not need to know what shards they are interacting with or how to calculate gas fees. Koala Wallet handles all on the background so the entire experience is simple and seamless.