Ask around and find out what gases, vapors, and/or fumes, in the atmosphere are the result of advanced technology. Then use a spectroscopy to look for those compounds in the atmospheres of earth like exo-planets. Who is doing this?
The best method of analyzing exoplanet atmospheres is called Transit Absorption Spectroscopy. First, we analyze the spectrum of an exoplanet's star. Then, we wait for the planet to pass in front of the star. When it does so, some of the star's light is absorbed by the planet's atmosphere, changing its spectrum. Analyzing this change will tell us what chemicals are present in the planet's atmosphere.
In order to detect life, there are several compounds which researchers look for. One of these is Oxygen. Oxygen is a highly reactive element, and will disappear from a planet's atmosphere very rapidly if it is not being produced anywhere. Life uses oxygen because it is so reactive, so if oxygen is present in a planet's atmosphere, it is a strong indicator of the presence of life.
In order to look for intelligent, technologically advanced life, researchers search for compounds which are not present in nature. One of the most notable compounds is the family of chlorofluorocarbons (CFC's). CFC's have no known natural source, and have a highly noticeable infrared absorption line.
Most of the absorption lines produced by these gases are in the infrared. This means that you need a special infrared telescope to observe them. Unfortunately, since our atmosphere is opaque at some of the frequencies required to find CFC's, you need to use space-based telescopes. There are hopes that the James Webb Space Telescope and the Transit Exoplanet Survey Satellite will be able to do this work when they launch. As this paper shows, the JWST would have to focus on a planet for ~5 hours to determine the atmosphere's composition.
This work can be done from the ground, but it is more difficult and has more uncertainty.