In order to answer this question we need look at the differences in these 2 types of stretching, let me explain.
Static stretching - is referring to when you are trying to increase the ROM (range of motion), an example would be a 1 leg hamstring stretch, as you sit and have 1 leg extended on the ground you would for a period of 30 seconds pull on your toes or ankle in order to stretch your hamstring, in other words increase the ROM, as oppose to,
Dynamic stretching - which is referring to when your body does the work for you, which in our case with the hamstring stretch, you would stand with 1 leg off the ground and you would swing that leg back and forward till you see that your hamstring has reached its fullest "current ROM".
I have quoted "current ROM" because that is the main difference between Static and Dynamic stretching, in the case static stretching as in the 1 leg hamstring stretch, within 10-15 seconds as you pull you would have reached your fullest "current ROM" whatever you do afterwards is trying to go beyond that, in Dynamic stretching all you are trying to achieve is your fullest "current ROM", in fact you would hear a lot of times trainers emphasize while doing Dynamic stretches not to over stretch, now that we know all that I will try to go over the reason for stretching and the effect it has.
Reason - The main reason for stretching is that while you are working out may it be in a strength program doing a bench press or a round house in a martial arts class, in both of these cases a certain muscle would have its ROM increased beyond its day to day ROM, in the bench press it would be chest (Pectoralis major), and in the round house kick it would primarily be your hamstring, having said that, not making sure before a workout that your body is ready to go to that ROM and usually because you have not warmed up enough can cause you to pull a muscle or even worst.
Effect - The effect stretching has on your muscles (besides for the obvious which is increasing ROM) is in fact counter productive, mainly in strength workouts and that is because muscles are at its strongest while they are contracted and tight in a bench press while your hands are extended upwards or in a bicep curl while your hands are curled up, so the more you increase the ROM the more your muscles are weakened and therefore you will be unable to fully perform while working out, and that is why there are people who don't stretch at all before a workout, now here is the big "revelation" this counter productive effect is mainly only in "static" stretching and that is because as we said in "dynamic" stretches you are not trying to increase your overall ROM.
I hope i have answered your question as to why Jillian wants you to avoid statically stretching before a workout, I know that there is a lot more to stretching for example hormone releases, but for its not for this article.
I should point out a few things.
static stretching is OK before your workout if don't pass the 10-15 second mark in other words don't try to go to the beyond, and it is common to do them as well as long its not over done, an example would be Tony Horton in P90X in which he does a combo of dynamic and static stretches before the workouts.
I'm not sure why Jillian is trying to avoid static stretches so strongly since to my knowledge her workouts are mainly cardio but I might be wrong on that, let me know DustinDavis.