Basically, a notice of abandonment is the trustee's way of saying that for one reason or another, the listed property cannot be liquidated for the benefit of your creditors. When a bankruptcy case is filed, real and personal property become part of a "bankruptcy estate" which the trustee is in control of. Once the trustee reviews the case and determines he cannot make any money for creditors by liquidating specific assets, they will file an abandonment notice on these items.
A Common Example
The most common reason the trustee abandons an asset is because it is burdensome to the estate. For example, if you have a car worth $10k but owe $15k on it, the trustee will abandon the car because if he were to sell it he would have to first pay off $15k to the vehicle lender. If the car is only worth $10k, there would be no money left over to pay any other creditors. The same is true for real estate. If you owe more than your home is worth, or if the costs of selling the home (ie. Realtor fees etc.) eat up any money available for creditors, then the trustee will abandon the asset back to you. Upon abandonment, the property is yours to do with what you choose and is no longer part of the bankruptcy process.