How is the amino group basic? According to the Khan Academy, the amino functional group is basic because it can remove $\ce{H+}$ from systems. Why is that?


Amino: Charged (forms $\ce{R-NH3+})$ at the pH of most biological systems. Since amino groups can remove $\ce{H+}$ from solution, they are considered basic.

My current understanding:

The $\ce{H+}$ atoms bond in a polar covalent way with $\ce{N-}$. So that means the amino acid does not release that many $\ce{H+}$ ions. I can see why this is not acidic.
$\ce{N-}$ also takes electrons from $\ce{R}$, which makes $\ce{R}$ become $\ce{R+}$. This should repel $\ce{H+}$ from elsewhere in solutions.

How does this functional group remove $\ce{H+}$ from its solutions?
Polarity and elementary electrostatics have nothing to do with acid-base definitions. Sometimes we try to find a reason, using our current understanding, and that ensues the danger of over-rationalization. At a truly fundamental level perhaps no chemist can tell you why an amine can accept another proton, may be someone can do quantum mechanical calculations and say why. So avoid over-rationalization... accept some observations as facts (temporarily until we develop a full understanding). 

One can titrate an amine with an acid, by this experimentally observed fact, exactly just like we titrate NaOH with HCl. So an amine group is indeed a base or say it behaves like base. The acidic proton cannot annihilate itself during a titration, it has to go somewhere. The amine group gets protonated, in return.