Why does 1/4 wavelength have a ground plane and 1/2 wavelength needs none?
I can't understand why we need to use a ground plane at 1/4 wavelength
  and 1/2 wavelength not needed.


A 1/2 \$\lambda\$ dipole has voltage and current waves like this: -



Picture from Wikipedia.

Now, if you focus your eye at dead centre of the picture you will see that the voltage is always zero volts. This is because a dipole is optimally driven with a balanced voltage source (\$V_O\$). A balanced voltage source is preferred for a dipole antenna. In fact, the voltage and electric field is zero all along the length of the green line below: -



This means you can optionally regard that green line as earth (providing the antenna is driven in a balanced way). Now if you were to cut the above picture in half you'd have a 1/4 \$\lambda\$ monopole driven with an unbalanced voltage source. An unbalanced voltage source is one that has typically 0 volts on one leg while the other leg does the voltage driving: -



And, not surprisingly, it has one half of the impedance presented by the half wave dipole. But, to keep the same radiation pattern you need to "force" an earth plane that does what the green line does.