Garage Sales
The purpose of a garage sale isn't typically to give items away free, but if a vendor is desperate to get rid of his possessions, he will often consider giving them away. Finding a vendor giving PS1 games away free can be challenging, but if you purchase another item from him, ask if he'll consider throwing in one or more PS1 games free.
Online
Most large cities have online classified sites in which residents buy, sell and trade goods. Many of these sites also feature sections for people to list items free; the concession is usually that to get the item free, you have to pick it up yourself. Peruse your city's online classified sites on a regular basis to see if anyone lists PS1 games for free. If you notice a listing, contact the person immediately to arrange pickup.
Friends
If you have friends who still own PS1 games but their PlayStation doesn't work, you might be able to convince them to give you the games with the agreement that they can visit your home to play the games with you. Conversely, consider arranging a trade. If the friend has a PlayStation 3, for example, offer to trade a PS3 game that you seldom play for several of the friend's PS1 games.
PS1 Emulator
A different way to obtain PS1 games for free is by downloading a PlayStation emulator and several games off the Internet. Emulators mimic video gaming systems, allowing you to play any game you desire for free. Many sites, such as Emulator Zone, ePSXe and PHP-Nuke offer free PS1 emulators for download.