Like everything else in pronunciation, it's really just something you have to practice, and also something you don't really have to worry about perfecting immediately. Since you can't.
But broadly, it's important to realize that in the dialect usually taught, a lot of certain vowel sounds in certain positions in a word become devoiced. This is why 'desu' tends to sound something like 'des', and why 'watashi' ends up sounding like 'watash' in 'watashitachi'.
In shitsurei, the 'i' sound in 'shi' and the 'u' sound in 'tsu' are pretty deemphasized. The result is a kindof rolling action with your tongue as you start the 'shi' and then glide over the 'tsu' directly into the 're-i'. Despite Japanese usually having a rather regular beat, this makes shitsurei seem to be said rather quickly (and I think it actually is).
I thought it was a pretty unique sound when I first heard it, that I didn't think was necessarily intuitive from the rest of Japanese pronunciation. It seems to follow in any 'shi-tsu-re' (possibly any shi-tsu-r-) words, though. For instance, I notice it in shitsuren, 失恋, to a lesser degree.
Well, all that's if you're having trouble trying to say it in a natural way, which I don't know if they do or not in that lesson (I'm not listening). If you're having trouble with a slowed-down pronunciation, I guess you're probably just having trouble with the 'tsu' sound, in which case.. practice
Anyway, someone come correct me with something more accurate. The amount of time I spend on pronunciation is not ideal.