How to get the style value using attr

This is what you are looking for(this code works):

$("#bgcolor").mouseleave(function(){
    var bodyColor = $(this).attr("style");
    $("body").attr("style", bodyColor);
});

But, instead of using attr you should use css, so the code will be:

$("#bgcolor").mouseleave(function(){
    var bodyColor = $(this).css("background-color");
    $("body").css("background-color", bodyColor);
});

JSFiddle

Check out the documentation for .css: http://api.jquery.com/css/

var bodyColor = $(this).css("backgroundColor");

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