Single entry bookkeeping is essentially a list of your income/expenses. A list of this nature means you can't be certain you got everything, if an item is missing or has been double counted it wont be obvious at first glance that it is wrong.

Double Entry bookkeeping adds a layer to your lists to ensure that everything has been accounted for, you are creating the same list but also tracking the opposite side which is what it affected.

The most simple case here is that the income/expense also affects your bank account. By tracking the bank account you now can check that your "Predicted" bank balance according to your list is actually what is in the bank account. If they don't match then you know for certain something got missed.

It doesn't solve all the errors that can possibly occur, but it makes sure that the basics are at least covered.