When you get a chance, let me know what you think if there are any signs that we are doing something new/different based on that video, I'm curious 
PoP gave a better answer than I ever could. I've studied coaching just to understand a little just so I can pick up bits and pieces coaches are trying to integrate into their game - but I looked more at the tactical work. PoP has actually done it this professionally though so his insight is far more valuable.
I would say anybody wanting to watch the video just skips the first 10 minutes. It's just some physical work, more of a stretch and warm up routine than anything else. I'll try to break little bits down and elaborate though.
The point PoP mentions when Klopp steps in is interesting. I think he is also mentioning awareness there "where are my mates". Then pointing with his fingers to show the limits of your peripheral vision - 120
o. And given that it is two touch - you cannot afford to receive, look then play (Emre special). So he wants people looking constantly so before the ball comes to them then have the layout of things in their heads and know where is safe to play or not.
I also think he is talking about playing on the half turn here and not standing square to receive the ball which makes you easier to press. He is indicating he wants them in motion to receive and play. A good example of this is Gini btw who almost never receives with his back pointing towards the opponent goal - he is always moving like a pendulum in relation to who is on the ball from center out. This means players tracking him are then moving away from the center too - thus opening up spaces for vertical passes beyond him into the more forward players. But his movement off center in this manner means his body is rotating away from center opening up more of the pitch with his peripheral vision.
So using this as an example, you can see how moving off center and opening up his body shape means he can see most of the pitch and all the opponents. So when you watching players dropping off to collect the ball, look at whether they are "boxing someone in" with their movement and also how they position their shoulders. Dropping vertically on top of someone and with your shoulders square to your own goal tend to be two things coaches will tend to drill out of you. Although big target men, like Lukaku, will typically do this with the intent to back into and roll the marker who follows them. There is a good example of (I think) Chirivella doing this at about 22m40 - he looks over both shoulders, make a good space to receive and then quickly plays it to someone he knows is in space. (after that though he generally looks really poor in there compared to Fabinho & Keita before)
You also hear the coaching staff (Ljinders?) shouting "always think before you get the ball - what is the best solution"
Form the defending team perspective, you are seeing pressure and cover shadows. Players trying to move so they are pressing from a direction that also blocks a passing angle for the ball carrier. So if you imagine if you have passing angles in front of you at 9 oclock and 3 oclock, pressing the ball carrier from 12oclock doesn't cause him much problems. However, pressing from 2 oclock means 9oclock is his only passing open that seems safe. Plus with this action you are simultaneously marking the player at 3oclock and the ball carrier which gives you are free player in defence. Plus by limiting his options, you can create a passing trap. A good example of us doing this is against Man City at Anfield where we created Fernandinho as the trap with Firmino in particular preventing vertical balls into him from the back which encouraged City to pass around Firmino and then into Fernandinho from wider angles. However Firmino doing this became the trigger that Fernandinho would get the ball within a few touches and we were already priming to ambush him from all sides when he did.
There is also the whole concept of compactness for transitions here at play too. Those on the outside, when the ball is lost need to move inside quickly to press and win it back. That creates the instinctive idea to compress space quickly. Likewise those inside, when ball is won back, move outside quickly to create space, passing angles to keep possession or, in games, launch the counter attacks. You here Pep talking about this around 21m. Then later he is saying "Open / Close boys" as in your shape in transitions.
One last thing is Fabinho being in the middle - the guy needing to receive passes from all directions and never has his back to a sideline which makes 360 awareness absolutely vital. I think this is very much to get him up to speed with receiving and playing under high pressure and intensity as quickly as possible given that Henderson likely misses the start of the season now. Then they switched him with Keita - same reason. He looks excellent in there too.
I think you also hear Ljinders shout at one point "Quick in mind, this is your idea. Control your body position to allow play one touch"
I also recommend trying to watch this without tracking the ball. Instead pick a group of players / space on the pitch to watch and then see how the players move - how good they are opening spaces, reacting, etc. I was watching Wilson for a bit and thought he was doing well in particular. He saw when a player was going to lose it for his team and quickly sprinted inside to back him so possession wasn't lost. He moved to create angles. Counter pressing he was shaping and pressing quickly in transitions. I also noticed Kent using quick skills to use players momentum to escape pressure very well.
