Why are primary carbocations less stable than secondary and tertiary ones?

Let’s say it the other way around. Tertiary carbocations are more stable than secondary and primary ones because they are stabilized by resonance and inductive effects.
When describing the stability of carbocations you must consider the inductive and effects of the ##R## groups which affect the charge on the ##C## atom.
Inductive Effects
An ##R## group is electron-releasing.
A carbon atom is slightly more electronegative than a hydrogen atom so an ##R## group has a slight negative ##^-## charge.
It can donate some of this electron density to the neighboring carbocation.
So the more alkyl groups you have on the cationic carbon the less its charge will be.
The cationic carbon will be at a lower energy and therefore more stable.
Tertiary carbocations have 3 ##R## groups secondary carbocations have 2 ##R## groups and the least stable primary carbocations have only 1 ##R## group.
Therefore a primary carbocation is less stable than secondary and tertiary ones.
Resonance Effects
The ##sp^3## orbital of a ##C-H## bond can overlap with the vacant ##p## orbital on the carbocation.
http://www.meritnation.com/img/shared/userimages/mn_images/image/1(68).png
This delocalizes the electron density in the ##C-H## bond onto the cationic carbon. Delocalization (resonance) lowers the energy of a system.
For a 1 cation like ##CH_3CH_2^+## we could write the resonance as:
##H-CH_2-CH_2^+ H^+ CH_2=CH_2 2 other contributors##.
The ##CH_3## group has three hyperconjugation contributors one for each hydrogen atom.
The ##(CH_3)_2CH^+## cation has six contributors and ##(CH_3)_3C^+## has nine resonance contributors.
The more resonance contributors the greater the stability.
So finally stability varies in the order