This means that someone who is experiencing abuse will probably not open up easily about what is happening. So, even if you've asked or told someone you are a safe person they can confide in, this doesn't mean that an abused person is going to take you up on that offer.
Some other reasons an abused person might be reluctant to open up:
1. Abusive relationships are more grown than 'simply happen', all involved parties gradually learn to tolerate slowly escalating levels of abuse over time. This means that this is a relationship with people who have invested large parts of themselves into the relationship that is now creating tension. All parties involved in any abuse based relationship have a lot to loose if things fall apart, so exposing a dangerous environment is very unlikely to occur without at least one extreme circumstance triggering some form of exposure.
2. It might be contrary to common perception, but people who experience abuse are actually extremely strong and capable persons. They've endured and survived things that most humans have not even thought were possible for a human to live through. People who experience abuse on a regular basis tend to find solutions to most issues they encounter without involving any outside intervention. They are much more likely to find a solution to ease the abusive environment without any external help, than they are to call someone who says "call me if you need anything".
3. The cycle of abuse literally builds trust and then uses that trust as a weapon. The cycle of abuse requires rebuilding trust through a honeymoon phase and eventually striking out against that hope/trust in the relationship. Every time these strikes occur, it is painful for the victim and costs at minimum extensive amounts of emotional energy. It is quite natural for a victim to learn quickly the importance of being extremely cautious about allowing themselves to trust another person.
4. Due to the generally accepted idea that the only way to stop abuse is to permanently cut off relationship and leave an abuser, many victims feel very judged when they choose not to leave the moment abuse is exposed. This is even more pronounced if they've left and gone back to the relationship a number of times or when family/friends have expressed consequences when they returned to the abusive environment.
Never assume that your role in supporting an abuse victim is complete the moment you offer them support that they decline to accept.