Diagnosis
Yeah, the bamboo has a light sealant, but it's not waterproof. Spills and harsh cleaners will seep in, causing white stains, swelling, or dulling the finish.
How to Fix It
For daily dust, use a dry microfiber cloth. For smudges or light grime, dampen the cloth with just water, or a 50/50 mix of water and white vinegar. Wipe with the grain, then immediately dry with another cloth. That's it.
If That Didn't Work
For set-in stains (like water rings), you can very lightly buff the area with extra-fine (#0000) steel wool, following the grain. Then wipe clean and apply a tiny bit of beeswax or bamboo-specific furniture oil to re-hydrate the spot.
Heads Up
Never use window cleaner, ammonia, abrasive pads, or 'all-purpose' sprays. They will permanently damage the finish. Also, don't let wet cups or plants sit directly on it.