Silent running basically "hides" all of your head within your ship.. which means your ship gets hot - really hot (since the heat isn't being vented into space like normal).
Generating heat is a separate matter - you can manually turn off engines / shields (which tend to gen the most heat). If you turn those off, you are basically in silent running and don't need to turn on "silent running mode". Except if you do turn off engines and shields... well... you're just floating around on whatever inertia you had!
So there's 2 ways to sneak in:
1. Silent running MODE: you turn it on, build up heat and try to get into the station quickly before you cook internally (using heat sink dispensers as yabba suggested makes this really easy)
or
2. Use inertia to glide in while being ACTUALLY cold: point towards the station, engage up the speed (hope you're lined up correctly) and turn off assist, then turn off engines and shields manually. You'll start cooling down as you fly in. Turn those all on rapidly when you are really close to the station and you'll zoom in. This is really, really tough.
And keep in mind, even if they start scanning you, once you are IN the station, the scans are canceled (because they don't have line of sight anymore). I can usually get in via option 1 without any difficulty.