At the bridge end you pass the strings over the saddle and then through the holes in the bridge from the neck side. Then take a short length of each string from the exit to the hole and bring it back across the top of the bridge. Pass it under the string where it enters the hole and then back towards the bottom of the guitar. Pass the loose end under the part of each string that crosses the top of the bridge (once is usually enough but for the thinner strings you could do this twice. Now, the important bit! Pull the string tighter from the part that will go to the nut and ensure that the wrapped around part string is trapped just behind, i.e. not on top of the bridge. If you do this there will be no slippage. If that grip is behind the back edge of the bridge, it will almost certainly slip.
At the tuner end, usual procedure, ensuring that you loop the string back round so it is gripped by the wound part and that you have at least two or three windings on the roller.
I'm sure someone will find a photo that shows this much more simply visually and do a blicky to it!