Grief follows extremely mournful situations. Many people have different coping mechanisms for grief. But coping mechanisms may not always be a method of dealing with grief, but also a way of masking it. They're also a way of internalizing a feeling that cannot be outwardly expressed, or suppressed.
Each person may have a different coping mechanism for grief, but we all grieve and we all try to hide it. It should also be noted that grief is not a set amount after every scenario, grief can be over minor occurrences as well as major.
Oskar internalizes his grief by searching for what he has left of his father, and holding on tightly to that which he does have. A specific coping mechanism that he uses is repetition. He repeatedly listens to the messages that his father left on his phone, because it is another thing that he is holding onto. While this may not rid him of grief, it helps him cope with it.
What's particularly interesting about Oskar's case is the way that he hides this grief. Every time that he listens to the messages from his father, he nestles up in his closet, forming a cocoon of sorts. I believe that this makes him feel safe and protected, from anyone that might take away this feeling from him (i.e. his mother).
What is most important to consider, in my opinion, is what grief is worth holding onto for the sake of remembrance, and what grief is necessary to let go.