That's because it's a high frequency transformer not a standard low frequency transformer or a switched mode power supply. The mains is rectified and connected to an oscillator driving a ferrite core transformer. The output is a 20kHz to 100kHz squarewave modulated by the mains frequency.
The standard low frequency diodes in your bridge rectifier will struggle to rectifiy the high frequency waveform, will have high losses and might overheat. The linear regulator will have virtually no ripple rejection at higher frequencies so the output will be noisy.
There will be a relatively high voltage between the secondary relative to earth because there will be a lot of leakage, both through the transformer itself and the Y capacitors, assuming the latter are fitted.
I've never tried to use such a transformer to power anything but a light bulb and I suspect it's more trouble than it's worth. You'll need to replace the low frequency diodes with Schottky diodes, add lots of low ESR ceramic capacitors , ferrite beads and a choke to the DC side but I don't know if it'll be enough to get rid of all of the noise.
Here are a couple of links:
http://sound.westhost.com/lamps/dimmers.htmlhttp://skyline-bg.com/ref/13707.pdf