[SalesForce] How to get specific value from map in Lightning component

This is easily done in just pure JavaScript, if you were looking for the way to do it using the component tags, I'm not sure of a built in way to do that.

Since setting the property on your component should trigger a rerender, you should be able to put this code in your afterRender function like so.

({
    afterRender: function(component, theMap) {
        var element = component.getElement();
        var ul = document.createElement("ul");
        var li;
        var text;
        for(var prop in theMap) {
            li = document.createElement("li");
            text = document.createTextNode(prop + ": " + theMap[prop]);
            li.appendChild(text);
            ul.appendChild(li);
        }
        element.appendChild(ul);
        return this.superAfterRender();
    }
});

Looking for something different?

Mohit points out that right now you aren't binding the value of the ui:checkbox to a component attribute, but that's not of concern here.

Using your code copied and pasted into a new component, I cannot reproduce your error. The checkbox state changes and it reports to console. Is there other code in the mix here? Is that your code verbatim?

That being said, both the dev guide and your code follow a bit of an anti-pattern in the event handler. Instead of setting a findable aura:id and hard referencing that in your controller, your code is much more useful (and would handle a list view) if you use the event source instead. Example:

handleChanged : function(cmp,evt,helper){
    var checkbox = evt.getSource();
    console.log(checkbox.get("v.value"));
}

This will fine the component that fired the event.