Why would a country want to erect trade barriers such as tariffs or quotas?

Basically the main reason is to protect the internal market.

Say one county wants to develop its own car industry. Before the cars can be produced so cheaply that they can be competitive with cars produces elsewhere, some time will pass, so during this time the country might want to make sure that its own cars will be bought.

It can limit the influx of foreign cars by imposing these measures, thus protecting the local car production companies.

Although most countries still erect trade barriers, they also want to engage in international trade. Because of that they enter into trade agreements with other countries. The reason for trade barriers is to protect domestic jobs and in other words to protect local industries. When there is no barrier and quota all the products and services could be easily funneled into country.