Silent running means your power and heat output is reduced to the minimum
hence your shields going offline and starting to get hot as your not radiating it into space anymore (its your heat signiture people see on scanners so no heat output = not on scanners)
If you want to stay in silent running you will need to eject your heat
I belive you still can be seen on the radar but only as a unknow contact and cannot be locked on. You can use the screen on your right to shut down other ships systems to keep the heat from rising and once you are completly cold you become invisible to the scanners.
So, Silent running can be uses as a way to get unlocked from radar and hopfully can make an exit. There is a couple of videos on youtube about silent running and shows you what it does.
Silent running (i.e. Stealth) is when your ship closes all vents and disables shields so that you no longer show up on radar. The disadvantage to this is that you are building up heat inside your ship and need to remove it (i.e. Heat Sinks) by firing off your heat sinks. Heat Sink shots DO show up on radar so don't eat where you sleep kinda deal.
The other side effect of going silent is you have no shields. If a pilot spots you and attacks, you have no defense. Your shields also have to recharge and come back online again when you turn silent running off.
You can shut down some of your systems so that your ship generates less heat, to the point that it's possibly to stay 'silent' indefinitely without using heat sinks. Naturally, this can also leave you pretty helpless, so be careful what you shut down.