Why does potential energy of a moving charge inside a closed circuit increases?

But in an electric circuit, if the charges, for instance electron move towards positive terminal then it is moving under the influence of attractive force. So how could it gain potential energy.

Keep in mind that conventional current is defined as the flow of positive charge, not electrons. When a positive charge moves towards the positive terminal through a battery work is done by the battery resulting in the positive charge gaining electrical potential energy. The work done by the battery is the consequence of the conversion of chemical potential energy to electrical potential energy by the battery.

Hope this helps.

if the charges, for instance electron move towards positive terminal then it is moving under the influence of attractive force. So how could it gain potential energy.

You have this backwards. As the electrons move towards the positive terminal they are losing potential energy. This happens e.g. in a resistor. In a battery, where there is a gain in potential energy, the electrons are moving towards the negative terminal.

By the way, it is much better to not think of electron current in ordinary circuits but just use conventional current. This focus on electrons may be contributing to your confusion here. Also, as you go to EM you will find that the energy is in the fields, not the charges.