He's interpretting it as a "Cleave" attack. It makes sense, for the old Cleave ability...but this isn't Cleave.
You have struck the enemy with such force as to deal a killing blow. Just like chopping wood, when you get a clean cut the blade goes straight through without resistance. Using the momentum of that swing the character can then move and utilize that momentum for an extra strike. In a way, its bit like Action Surge in that you are getting a burst of action on top of your normal action.
How to RP it:
Blood-lust - if the fighter is particularly barbaric, he may see the damage he inflicted and give him a rush of adrenaline to quickly follow with another attack.
Cleaving/follow-through - Slashing entirely through an enemy and carrying onto a nearby target. Additionally, if this is used against the same target, you can see a downward slash, chopping the targets arm then leg.
Redirecting Momentum - Any broadly swung weapon could be seen as the critical hit as the attacker having such perfect timing as to recover from the attack much quicker than normal. For example, He was able to follow through from a high guard to a low guard perfectly after the attack, thus enabling a reverse swing much quicker than a blocked attack that made him return to high guard.
This is what I was looking for. I'm sure there's carry-over from prior dnd versions.