Why is methyl group more electron-donating than tert-butyl group?
Methyl group is not more electron donating than tert-butyl group, as the video provided in the link suggests (11:20 of the video), which is true.  

Inductive effect is a permanent effect which results in displacement of electrons along the chain of atoms in the molecule (here, carbon atoms).  The reason for this displacement can be worked out from the electronegativity difference between the atoms. 

Consider the tert-butyl group.  In a C-H bond, the electron density is directed towards carbon.  Methyl group, which has three hydrogen atoms attached to a carbon, pushes the electron density towards $\ce{C2}$ of tert-butyl acetic acid.  So do the two other methyl groups.  The amount of partial negative charge accumulated on $\ce{C2}$ of tert-butyl group (has 3 methyl groups) is more than that on methyl group (only has 3 hydrogens) of acetic acid.

The more negative charge on $\ce{C2}$ in tert-butyl acetic acid destabilizes the conjugate base, compared to a simple acetate ion.  Hence acetic acid has less pKa than tert-butyl acetic acid.