Why is second ionization enthalpy higher than first?

Hint: The ionization enthalpy is the amount of energy that is required to remove an electron from the isolated gaseous atom. To lose an electron the atom has to be supplied with energy. This energy is called the ionization enthalpy of an atom.

Complete answer:
When you remove an electron from the neutral atom, the atom gains a positive charge and becomes a positive ion, hence the name first ionization enthalpy.
After the first ionization enthalpy is supplied, the number of protons becomes more than the number of electrons. So the attraction between the protons and electrons increases. The nucleus pulls the electrons towards itself because it has more charge.
Now we need to understand the Second ionization enthalpy.
The amount of energy needed to form a dipositive ion from a unipositive ion by the removal of an electron is defined as a second ionization enthalpy.
When you have to remove the outermost electron from a positive ion, the proton and electrons are more tightly bound than it was before. So, to lose the electron from the last shell more energy is required as it becomes harder to remove the electron.
Thus, the second ionization enthalpy is greater than the first ionization enthalpy.

Note:
If you keep removing the electrons the positive charge increases which increases the attraction force which eventually decreases the size of the atom. But the nuclear charge on the atom keeps rising. Also, remember that the second Ionization enthalpy of Alkali metals is much more than their first ionization enthalpy.