Why do people pay tax, and why is it very important?

The simple answer to the question "why do people pay tax" is that willful failure to pay taxes according to the tax code will result in seizure of your back accounts, property, and maybe even imprisonment.

A simple example for if people paying no taxes is there won't be public schools. So kids get education through private schooling, home schooling or they just wonder outside and admire the beautiful blue sky everyday.

Seriously, most modern governments use taxes to fund welfare and public services. These services can include education systems, health care systems, pensions for the elderly, unemployment benefits, and public transportation. Energy, water and waste management systems are also common public utilities.

The Four "R"s
Taxation has four main purposes or effects: Revenue, Redistribution, Repricing, and Representation.

The main purpose is revenue: taxes raise money to spend on roads, schools and hospitals, and on more indirect government functions like good regulation or justice systems. This is the most widely known function.

A second is redistribution. Normally, this means transferring wealth from the richer sections of society to poorer sections, and this function is widely accepted in most democracies, although the extent to which this should happen is always controversial.

A third purpose of taxation is repricing. Taxes are levied to address externalities: tobacco is taxed, for example, to discourage smoking, and many people advocate policies such as implementing a carbon tax.

A fourth, consequential effect of taxation in its historical setting has been representation.[The American revolutionary slogan "no taxation without representation" implied this: rulers tax citizens, and citizens demand accountability from their rulers as the other part of this bargain. Several studies have shown that direct taxation (such as income taxes) generates the greatest degree of accountability and better governance, while indirect taxation tends to have smaller effects.

People pay taxes to support the projects and programs of the government. Defense, education, Infastructure, etc. But, that doesn't mean that the current tax structure is at all important. The progressive income tax system is punitive to those people who work hard and earn enough to live on (or more). It is also punitive to businesses and business owners, even those people who only have a few employees. We need to take a look at a national sales tax system, which would essentially allow people to choose how much tax they pay, and is projected to bring in MORE revenue than the current system.
Einstein said "The hardest thing in the world to understand is the income tax" - he wasn't talking about the forms.