It's really cool. the "relay" is the part that makes it flash. the way old relays work is there's a strip of 2 dissimilar metals back to back that have different expansions as they heat up. as power goes through the strip, resistance makes it heat up, but the uneven expansion f the 2 metals makes it bend. as it bends, it 'pop' clicks away, breaking contact. now without power through it, it cools until 'pop' it clicks back and makes contact again.
repeat repeat it makes the characteristic tick tic tic of a blinker, and continuously flashes the blinker lights on and off.
armed with that info, you should recognize it requires a certain load for the strip to heat up and click on and off. that load is created by the original bulbs. if you use a lower wattage bulb, such as LED's, or if one of the bulbs burns out, it can make the relay either not flash, or likelier get more power through the relay so it heats and flashes extra fast.
so that's what's going on. what's the solution then? Here for LED's you're going to need an electronic relay. those have circuits that create the flashing, as opposed to relying on the physical clicking of a bimetallic strip, and thus arent sensitive to the drastic swing in wattage introduced by using LED bulbs. viola!