An object, at its simplest, is just a collection of state and functions that operate on that state.  A closure is also a collection of state and a function that operates on that state.
Let's say I call a function that takes a callback.  In this callback, I need to operate on some state known before the function call.  I can create an object that embodies this state ("fields") and contains a member function ("method") that performs as the callback.  Or, I could take the quick and easy ("poor man's") route and create a closure.
As an object:
class CallbackState{
    object state;
    public CallbackState(object state){this.state = state;}
    public void Callback(){
        // do something with state
    }
}
void Foo(){
    object state = GenerateState();
    CallbackState callback = new CallbackState(state);
    PerformOperation(callback.Callback);
}
This is pseudo-C#, but is similar in concept to other OO languages.  As you can see, there's a fair amount of boilerplate involved with the callback class to manage the state.  This would be much simpler using a closure:
void Foo(){
    object state = GenerateState();
    PerformOperation(()=>{/*do something with state*/});
}
This is a lambda (again, in C# syntax, but the concept is similar in other languages that support closures) that gives us all the capabilities of the class, without having to write, use, and maintain a separate class.
You'll also hear the corollary: "objects are a poor man's closure".  If I can't or won't take advantage of closures, then I am forced to do their work using objects, as in my first example.  Although objects provide more functionality, closures are often a better choice where a closure will work, for the reasons already stated.
Hence, a poor man without objects can often get the job done with closures, and a poor man without closures can get the job done using objects.  A rich man has both and uses the right one for each job.