The best explanation for local self-government is democratic decentralisation, which refers to the process of transferring power and decision-making from central or national governments to local or regional governments.

Democratic decentralisation allows for local authorities to have greater control over their communities and allows for citizens to be more involved in the decision-making process. This approach is based on the principle of subsidiarity, which means that decisions should be made at the lowest possible level of government that is capable of making them.

Local self-government also provides a sense of ownership and responsibility for the community, which can lead to more effective and efficient governance. By involving citizens in decision-making, it can help to ensure that policies and programs are tailored to the unique needs of the community.

Federalism is another form of decentralisation that divides power between a central government and smaller regional governments, but it does not necessarily involve a direct transfer of power to local communities. Administrative delegation involves the transfer of administrative tasks or functions to local authorities, but it does not necessarily involve decision-making power.

Direct democracy refers to a system where citizens directly vote on policy decisions, bypassing elected representatives. While this can be a form of decentralisation, it is not necessarily related to local self-government.

In summary, local self-government is best explained as an exercise in democratic decentralisation because it involves transferring power and decision-making to local authorities, allowing for greater citizen involvement, and promoting effective and efficient governance.