The methyl group carbon has a partial negative charge due to the polarity present in the C-H bonds. This partial negative charge is preferentially donated, leading the methyl group to have the +I effect.
An important way to consider the stability of a tertiary carbocation is through the process of hyperconjugation. Neighboring C-H or C-C bonds, parallel to the vacant p-orbital of a carbocation, can donate some of their electron density to the empty p-orbital, thus stabilizing the positive charge. Tertiary carbocations have a greater capability of providing this electron density, and thus are more stable.
The positive charge present on the nitrogen in either the -NR3+ or -NH3+ groups induces the extent of the -I effect exhibited by both groups. Although it may seem that the presence of alkyl groups would lend a greater +I effect to the -NR3+ group, there are other factors that result in this "anomaly" of sorts.
The factor that causes this is the issue of steric hindrance. The alkyl groups surrounding the nitrogen are relatively large and do not want to be in the vicinity of each other. Although the structure of the -NR3+ group would appear to be tetrahedral, as the alkyl groups push away from each other, they move towards what seems to be more of a planar structure. This is an increase in bond angle from the tetrahedral structure.
It is important to note that the greater the bond angle, the greater %s character. S orbitals are typically closer to the nucleus as compared to the p orbitals, and thus, the electrons are held closer -- the electronegativity of the nitrogen atom will be greater than if it was in the -NH3+ group. As such, it exhibits a greater -I effect.