Andrew Smith is a 3rd-degree BJJ black belt based out of Richmond, VA (Revolution BJJ). He runs BJJ Path, a video tutorial website.
Certain moves in Brazilian Jiu Jitsu are "must know" moves. The kimura from side control is surely in this category; matches in sport jiu jitsu are won with this technique from white belt all the way through the highest level black belt competitions, and every day at the gym. Here's a straightforward breakdown and examination of the set up and finish behind this fundamental, essential submission.
Starting from head and arm side control (with your right arm underhooking their arm or controlling their hip low, and your left arm underneath their head), pass your left arm over to the other side of your partner's head. The idea here is to pinch with your left elbow, trapping just above your partner's left elbow, taking care to ensure that you are controlling their arm at their triceps, not over their shoulder. Next, switch your hips to reverse kesa gatame, making sure to use your left arm to block your partner's knee (even in rolling, it's pretty easy to get clocked in the face by a knee!). Isolate their "inside arm" (their left arm in this case) by scraping your hips along the ground and walking them up to your partner's armpit, rendering their most common side control escapes all but useless. Now, thread your left arm through your partner's arm, with your hand beginning to come through on the other side, just above your partner's forearm. Next, grip their wrist with your right hand while grabbing your own right wrist with your left hand. Note: try to use "all five fingers" together, including your thumb, creating a hook of sorts to pull the kimura inward. Drive their hand away from their gi (and away from anything else they might try to grab for no-gi), making a right angle bend in their arm, with their arm away from their body. Finally, switch to modified kesa gatame so that you can step over their head, keeping them from following you as you begin to torque their wrist backward.
A second approach to snag the Kimura from side control is a more reactive approach: that is, allowing your partner to move first, setting a trap for them. Start with 100 Kilos Position, keeping enough weight on your partner so that they have little choice but to go for an underhook and to try to turn in toward you. Once they begin this movement, allow them to get so far as to reach around your waist or leg. While they are grabbing for this, switch your hips so that you can drive them back flat with your right foot. Simultaneously, reach over them with your left arm, once again looking to thread the hand through (the "key" for the kimura). Once you connect your hands with the kimura grip, you can use this in conjunction with the "driver" leg to flatten your partner out. Separate the arm from the body, switch your hips, and step over the head, once again, to get the submission.
Keep in mind that any time you can get your partner's limbs away from their body, the weaker they generally are. This is a good reason to use a frame to help accomplish this: drop your elbow beside your partner's hips, effectively making your forearm a solid frame so that your partner won't be able to get their hand back in close to their body. You can also see the concept of blocking their head well from this angle; make sure that, as you are torquing their shoulder by first lifting their elbow, then dragging the wrist back, if needed, you are blocking their ability to sit up and follow you by stepping over the head. Note that it's not about hooking close to their head, but rather about inhibiting their ability to follow you by sitting up. This means dropping your knee closer to their head.
The kimura from side control must be one of the all-time most used submissions across all platforms where the submission is legal. It's not only high percentage for white belts, but even black belts have a tough time defending if the set up is done properly. These entries are merely the tip of the iceberg as far as kimura set ups go, and there are dozens more options to finish as well. Start with the basics, and all things will evolve over time!