Double cleansing is the Korean approach to evening cleansing: an oil cleanser first to dissolve oil-based impurities (makeup, SPF, sebum, pollution that's stuck to oil), then a water-based cleanser to remove anything water-based (sweat, dirt, residue from the first cleanse).
Single cleansing often isn't enough to get everything off, especially if you wear daily SPF (which you should) or makeup. What gets left behind can clog pores, dull your complexion, and stop your serums and moisturisers from absorbing properly.
Double cleansing matters most in the evening. It's especially important if you wear heavy or long-wear makeup, mineral-based SPF, or live in a high-pollution city. In the morning, a single water cleanse is usually enough.
The Korean approach works because each cleanser is doing a specific job rather than one cleanser trying to do everything (and either failing or being too harsh).