CSS3 animation transition: opacity not working

You can just use visibility without using display:

.top-nav li ul.drop-down {
    list-style-type: none;
    position: absolute;
    top: 32px;
    left: -40px;
    visibility: hidden; 
    opacity: 0;
    -webkit-transition: opacity 0.3s;
    -moz-transition: opacity 0.3s;
    -o-transition: opacity 0.3s;
    transition: opacity 0.3s;
}

.top-nav li:hover ul.drop-down {    
    visibility: visible;
    opacity: 1; 
}

Demo.