"Spider guard" is so-called because your legs and arms retract and extend at roughly equal or seemingly random intervals, making you look like a spider devouring its prey. An equally good (or possibly better) name might be "puppet master guard", because you are able to play your partner like a real-life puppet with spider guard. It's one of the types of open guard that has been around for the longest in sport BJJ, and although how the guard works is relatively well understood across a wide range of skill levels, not everyone has a systematic approach to this game. We're going to enter to spider guard and take a look at a couple of basic options depending on your partner's reaction, and adjust accordingly.
Initially, it might be easiest (or, at least, most comfortable) to enter into spider guard from the closed guard. Start with J-hooks on both sleeves, but with your knuckles on top of your partner's forearms, right where they meet their hands. This will help with pushing as well as pulling. Hip out to your right (make sure your butt moves to your right) as you push your parnter's left sleeve away from their body, creating enough space for you to slide your right knee across their midsection, almost as though you are going for a basic scissor sweep. As soon as you have the space to do so, open your right knee up, putting immediate pressure on your partner's left biceps. Now repeat the process on the other side, sliding your left knee through just enough to be able to open up your knee. Now you have a fantastic defensive guard, with inside control on both sides, and with double sleeve control. Finally, from here, just hip out enough so that you can step on your partner's left biceps with your right foot (you will need to hip out to the left as you do this), and then recenter yourself under your partner.
They key to the next several positions (including those taught in future tutorials) is to step on their biceps with one side (here, it's your right foot on their left biceps every time), and then to think of this as your constant, not letting go until it's time to finish whatever action you're taking (sweep or submission). Once you have this grip, you can clear your partner's inevitable pants grip on your left leg by pulling in and kicking out, and then swim your left knee over the top of your partner's left arm. Next, make a huge counterclockwise circle with your left leg, ultimately building momentum so that you can pull your partner forward. Once they are light, complete the circular motion by chopping out (or blocking) the bottom of their right leg, completing a "scissor sweep" type motion.
One type of resistance you are likely to encounter is your partner stepping up with their left leg, ultimately preventing the spider scissor sweep. Not to worry: you can hip out to the side and insert your left butterfly hook into the crook of their knee. Note: there are two ways your partner can position their leg in order to avoid the earlier sweep. Once you have gotten your butterfly hook inserted, go for the sweep to your right with a classic steering wheel hand motion. If you miss the sweep but get close, you can also catch the triangle choke on their way back up. If their leg is in the middle, you can kick it out and hit the triangle there as well, although this one takes some timing and repetitions to start using during rolling.
While there's a lot more to running a good spider guard game, this is a good place to start. Entering from the closed guard will give you the confidence you need to try the positions right away. If you miss, you can often go right back to the old familiar home base of closed guard. Try out the basic spider scissor sweep as a go-to option, and then note what your partner does to prevent being swept. Ultimately, you're going to learn (or figure out) how to get around this resistance and sweep or submit them anyway.