How exactly do they heal then if it isnt a spell?
Good question.
Wounds don't have to disappear for their effect to be mitigated, and not all damage is physical.
Wounds themselves aren't usually what kill, it's the resultant loss of homeostasis (primarily, inability to maintain blood pressure/circulation). If a wound stops bleeding, its effect is largely nullified (which is why First Aid - Self Aid/Buddy Care in the military - saves lives and even gets people back into the fight).
Wounds are not
only cuts. They can also be severe hematomas (bruises) or just intense pain (which can also be life-threatening - pain leading to hypovolemic shock and loss of homeostasis), among other things. But this specificity is not part of the game.
The D&D definition of Hit Points is
"a combination of physical and mental durability, the will to live, and luck."
The mechanics of the game do not go into a granular level of what kind of damage is taken. Any mechanics that do are optional and outside the conceits of the game. D&D Hit Points are
abstract - and they were defined that way for a reason.
However, for verisimilitude purposes, Inspirational Hit Point recovery can be explained as stimulating a sympathetic nervous system response. That response triggers adrenaline, norepinephrine, and serotonin.
Adrenaline and norepinephrine raise blood pressure, heart rate, and increase release of glucose to the muscles (physical durability).
Seratonin immediately raises mood, promotes blood clotting, and also assists with raising blood pressure (mental durability, will to live, and physical durability).
In other words, Inspirational Hit Point Recovery gives your mind and body a reason to fight, and augments functions of the body to regain homeostasis long enough - possibly - for the body to maintain homeostasis after the spike in neurotransmitters has worn off.
Warlord Inspirational Hit Point Recovery is consistent with the games definition of Hit Points, and consistent with the logic behind Hit Point recovery due to Short and Long Rests. (If Inspirational Hit Point Recovery doesn't affect wounds, how then does taking a nap affect wounds?)
Warlord Inspirational Hit Point Recovery is also consistent with fictional tropes. D&D is a system replete with fictional tropes hard baked into the system. Inspirational Recovery is no less or more rational than the tropes already incorporated into the game.
And, Warlord Inspirational Hit Point Recovery is even consistent with some real-world examples.
Conclusion: There is no rational reason for Inspirational Hit Point Recovery to not be in the game. It is a rational reason for personal choice on whether to use it or not - but not to keep it out of the game for others.