How do electric charges exert forces?

Hint: First we need to understand that any force, be it gravitational, electrostatic or even magnetic that is experienced by particles of the same nature is a result of a field generated by these particles. That is if we say two charged particles are kept close to each other, then they experience electrostatic force as a result of interaction of them with the other particle’s electric field.

Complete answer:
To explain it in detail, let us assume an electric charge (q) in space. Now, this electrical charge will create an electrical environment around it which we call an electric field. Now let us assume there are other charges in the region. Then, this field interacts with the other charges in the region and exerts force on them. Thus, interaction between the charges can be described as a two-step process, that is creation of the field and interaction of this field with other charges in space.
Mathematically, Coulomb proved that the force exerted by any two charges on each other is directly proportional to their product and inversely proportional to the square of their distance. This implies that greater the charge, greater the force of attraction or repulsion. Also, the magnitude of this force decreases with increase in distance.
Thus, Coulomb’s law of Electrostatics for two charges (${{q}_{1}}$ ) and (${{q}_{2}}$ ) at a separation (r) can be written as:
$\Rightarrow {{F}_{electrostatic}}=k\left( \dfrac{{{q}_{1}}{{q}_{2}}}{{{r}^{2}}} \right)$
Where, $(k)$ is the constant of proportionality whose value is $(9\times {{10}^{9}}\dfrac{N.{{m}^{2}}}{{{C}^{2}}})$
Thus, we see that electric charges apply force on each other by interacting with each other in their Electric field of influence.

Note:
There is a striking similarity between the force of gravitation and the electrostatic force. Both these interactions depend on the product of their particle’s parameter, that is mass for gravitation and charge for electrostatic force, and both are inversely proportional to the square of the distance between the particles. There is one big difference. And that is, the gravitational force is always attracting in nature as there is no concept of negative mass whereas electrostatic forces can be of attraction as well as repulsion.