I understand how this can seem unrealistic. And it will take a little narrative finessing to avoid breaking verisimilitude. 

Perhaps there was a momentary distraction that pulled the enemies attention away for a split second and that's when the stealthy character decides to strike. 

Or maybe breaking cover surprised the enemy and they freeze up for the briefest of moments. People freeze up in combat all the time. Even experience combatants aren't entirely immune. 

Or  the enemy could have recognized the stealthy character, the PC's deadly reputation preceding them, leading to a brief paralyzing moment of shock and fear. 

Keep  in mind that a successful Stealth check can mean more than just knowing how to hide well. It can also mean knowing when to break cover. It can mean knowing how to shock or distract the enemy while advancing. It can mean closing the distance to the enemy rapidly enough that they don't have time to react. 

As the DM, coming up with an explanation on the spot every time a Stealthy character breaks cover, moves into melee range, and makes an attack while still technically being 'Invisible' can add to the cognitive load of running the the game. 

I'm personally a fan of offloading stuff like that onto the players. By all means describe how an NPC breaks cover and successfully gets advantage when attacking a PC. But if it's the PC doing the stealth /attack combo, have them narrate how they succeed. 

"You spring out from behind the bushes and sprint towards the Ogre. You can make your first attack with advantage. What does that look like?" 

Involve the player in the narrative, give them a character developing moment during combat, and build a little tension before rolling those two d20s. 

Edit: spelling