So lactic acid or lactate is produced when there's not enough oxygen around to break glucose down aerobically. But contrary to what many believe, lactate doesn't produce the pain we experience during intervals or the next day. The real culprit are hydrogen ions, also a biproduct of anaerobic metabolism, which essentially are changing the pH in the muscle cells which they don't like. Muscles, and our entire body really, function in a very narrow range of pH.
The reason we thought it was the lactate was that it was the most easily measured chemical, but the hydrogen ions actually were elevating in sequence with the lactate. We actually use the lactate later on to produce energy through another pathway which was why many physiologists began to question that lactate would cause discomfort....it didn't sit well that an energy source would be inhibitory in any way.
So when you improve with interval workouts what you're actually doing is enhancing your muscles' ability to buffer against changes in pH.
It's still fine to refer to lactate threshold though since we still can use lactate as a metric of the muscles acidity.
I tell my athletes to think of their muscles as a bucket with a hole in the bottom. As we run fast we are dumping more acid into that bucket....producing more and more hydrogen ions (and lactate), and the hole in the bottom is draining that acid away. As we train, we make that hole larger and larger so that it can drain more acid.
Hope this helps.....