It's a pawn shop that deals specifically in tickets and gift cards, and its cornerstone items are transit fares--shinkansen, local trains, and long distance buses. Usually located next to major railway stations in cities, this shop is an easy way to save a few bucks whenever you're planning to travel. They are usually tiny storefronts with a single counter and often-handwritten signage you mistake for a travel agency: dozens of destination cities listed up and down the storefront along with the (shinkansen) ticket prices to those cities.
The selling rate is usually 85~95% of the face value of the ticket. For bullet trains, that can be 1000 yen off, and the shops are often located just a couple minutes' walk away from the JR wickets.
Being a pawnbroker, the kinken shop also buys. Bought a shinkansen ticket but your plans were cancelled? Sell it off. For travel, this is particularly relevant to 回数券 (kaisuuken), or ticket books purchased to travel on the same route several times. If some of the tickets in a ticket book are used up, the remainder usually can't be taken back to the station for a refund. But the kinken shop will buy them off you.
Aside from transit fares, kinken shops usually deal in other cash-value items like gift cards and postage stamps. If you mail home frequently, for example, you can buy your stamps here for cheap and save a few yen on each. Or if you are planning to make a big purchase at a local department store, you can stop by the kinken shop first and pick up an applicable gift card for about 90% of the face value.
Some kinken shops also branch out into other items that hold value well, like bottles of perfume and precious metals. Foreign currency exchange is also sometimes offered, though the rates usually aren't very competitive.
If you're unsure where to start looking in your city, start online. A simple search for 金券ショップ on Google Maps will let you know where you can find one of these shops in your area.