Whether you're looking at building muscle, losing fat and/or increasing fitness, correct eating is essential to achieving that goal.
I have heard theories just like the one you have quoted. There is also the idea that resistance training on an empty stomach encourages testosterone production, thus encouraging muscle gain. Another concept with cardiovascular exercise is to work on an empty stomach in order to 'tap into' your fat stores. Personally I'm not a believer in any of these theories for a number of reasons.
By not eating prior to your workout, your body will be in a starved state. If your body cannot access the right nutrients in order to perform the desired exercise load, guess what's going to happen? Intensity suffers - period. There's a tonne of articles on Amino Z as to why intensity is a key factor in assisting fat loss, muscle gain or fitness improvements.
Meanwhile let's assume you complete your workout. Your body is now in a state of catabolism - it's breaking down muscle and fat tissue in order to recover properly. Again if you don't feed your body, you are starving it of essential nutrients required for effective recovery. This will inhibit protein synthesis (ie. building muscle), slow down your metabolism (thus slowing fat loss) and run your body down significantly (decreasing fitness improvements amongst other things).
In my opinion, one of the most important meals of the day is IMMEDIATELY after your workout (post workout). I recommend this to all of my clients. A serving of fast acting protein (eg. whey protein isolate) and carbohydrate (eg. dextrose) is an essential tool for getting your body back on track to build muscle, lose fat and/or increase fitness. I also recommend a pre-workout meal or shake in order to allow your body to perform at it's peak during the workout.