Finally had time to start reviewing. Here goes:
1) I recommend at least naming the other core rulebooks, though it would be better to give a quick summary of each. At least as far as Tech Manual and TacOps (both volumes).
2) In the intro section, I think a mention about simultaneity would be helpful, especially since you use the phrase "we all take a turn". You do get to it around minute 3, but I think that's a bit late, and could cause confusion. I would also recommend using legal moves as examples, even this early. This will prevent confusion later. I also see you're using tanks in your examples. You may want to stick to 'mechs at first.
3) It might be a little early to say "expending movement points", and it's not exactly clear with regard to where you start. I think you're saying a unit starts one hex off the board, and needs to move into the first hex, vice starting there.
4) You might want to mention "level 0" flat ground in your initial list of terrain. This video feels more like a miniatures discussion than rules, and I'm not sure it fits here.
5) The text note about jumping is a bit jarring at the beginning. I think that might have been better accomplished with a verbal "... we'll start with just walking and running...". The point about bipedal 'mechs really should include the words "without turning". When you do get to turning, you say "change hex side" instead of "change facing". I recommend the latter. Also the "about face" section included walking forward one hex for 4 MP... the "about face" itself only took 3 MP. I would also recommend a remark at the beginning that the various movement modes impact firing. It's interesting they recommend using Black for running in the book. My group uses White for Walking, Red for Running, and Blue for Jumping. We set the die to 5 if the unit moved for zero, and 6 (as recommended) if held still, but the color still matches the movement mode (usually White, for obvious reasons). In general, when you switch to a text page, you should already have it zoomed to a readable level. There's really no reason to make your viewers wait for you to fiddle with it. Having just finished episode 5, I have to say it seems a bit late to be introducing the tables, and a full 10 minute episode seems a little long to go over movement basics that don't actually include jumping (the ultimate short cut).
Ok, that's the first third. I'll try to get at least another third knocked out tomorrow...