The details of how it looks in-game are not specified, so it's up to you.
The description of the Mask of the Wild trait that you've quoted in your question is all that the books state about how it works. The description only explains the mechanics - you can hide when lightly obscured by natural phenomena - so what that actually looks like to other creatures is up to you.
Given that it doesn't automatically cause you to be hidden (it just lets you try to hide even when not totally obscured), a logical interpretation might be that your wood elf character is better at camouflaging themselves and blending into the natural environment around them.
The Mask of the Wild racial trait is briefly addressed in the Sage Advice Compendium:
Do the lightfoot halfling and wood elf hiding racial traits allow them to hide while observed?
The lightfoot halfling and wood elf traits—Naturally Stealthy and Mask of the Wild—do allow members of those subraces to try to hide in their special circumstances even when observers are nearby. Normally, you can’t hide from someone if you’re in full view. A lightfoot halfling, though, can try to vanish behind a creature that is at least one size larger, and a wood elf can try to hide simply by being in heavy rain, mist, falling snow, foliage, or similar natural phenomena. It’s as if nature itself cloaks a wood elf from prying eyes—even eyes staring right at the elf! Both subraces are capable of hiding in situations unavailable to most other creatures, but neither subrace’s hiding attempt is assured of success; a Dexterity (Stealth) check is required as normal, and an observant foe might later spot a hidden halfling or elf: “I see you behind that guard, you tricksy halfling!”
As you can see, they're not automatically hidden, just able to hide where others wouldn't be able to be.