The bad news is you are only out of bronze in name. Your MMR will still be around the same level. So in Silver 5 you are generally still playing with players ranging from Bronze 2 - Silver 4. Maybe getting an outlier. This is also assuming that you've played a fair amount of ranked so your MMR is stable and not still in the settling stage.
Remember, your Tier/Division really only represents your end of season prize placing. Your MMR will determine who you play against. Your MMR goes up when you win, depending on the MMR of your opponents (tougher opponents == greater gains) and lowers the same way (tougher opponents == less losses).
As for how to not get placed there at the beginning of season 6. Just don't worry about that. It depends how hard they reset people, but if you get better during season 5, you will be able to climb back up to your MMR's tier range easily. Just concentrate on your play and getting better and you will be able to stay out of bronze MMR.