Braking while downshifting/Blipping the throttle
Blipping the throttle means to roll on quickly and for just an instant to get the engine speed to match the lower gear you're shifting into. This means matching revs between the rear tire and engine while downshifting.
Let's go back a bit. You are accelerating. You are in 5th gear and the engine is spinning really quickly, so you shift up to 6th gear. Notice how the rpms drop and you are no longer revving the engine so high.
Now, think backwards - think deceleration. Same engine, same rpms, just the other way around. You're in 6th gear and you want to go to 5th. You could just jam it into 5th gear, but you'll notice the tach shoot way up and the bike lurch a little bit. This is the same reason the rpms dropped when you went from 5th to 6th. The engine has to spin faster in a lower gear to make the rear tire spin at the same speed.
The fundamentals are that each gear is a different gear ratio. Let’s make up some numbers for illustration purposes. Pretend that 3rd gear at 10,000 rpm is 50 mph, and 4th gear 50 mph is 8000 rpm. So, we're riding along in 4th gear. We slow to 50mph and shift down. If we don't do anything, just slam it into 3rd gear, the engine has to rev up to 10,000 rpm. The rear wheel pushes the engine, basically. If the differential in revs is too great, the rear tire can chirp and the bike can dance about.
What we are trying to do is blip enough so that the engine will spin the rear tire at the same speed the rear tire is rolling along the ground. Get it right and it will be smooth. Too small or no blip and you could lose traction. This is a problem at higher rpms in particular. Slowly decelerating isn't the same thing. So, when slowing down quickly you can either feather the clutch or blip the throttle to match the rpms.
How to blip
Rev matching, or blipping as we're slowing down, all happens in a fraction of a second, like this:
So, pull in the clutch lever, give a quick small increase in throttle (it doesn't take much at all when there is no load on the engine) put the bike in the lower gear, and then ease off the clutch. Ease back off the throttle to resume deceleration.
This takes practice, and is really only needed when decelerating quickly. Once you get comfortable with it, you'll just do it naturally when you need to do it. It gets to the point where you don't even know you're doing it - and you can maintain an emergency threshold braking (with 2 fingers) on the front tire while blipping (with your thumb/palm).
Now, because we are braking, our two fingers are squeezing the brake lever. This leaves only your palm or thumb left to blip. Some like to raise their hand over the bar to leave room to blip with the inside of the thumb. Others will hold the bar normally and blip by pulling the palm over the throttle. Their wrist cocks and ends up below the bar. It doesn’t matter how you do it, so long as you maintain brake control.
Obviously, this is a technique that needs to be practiced on a lonely stretch of asphalt, or as lonely and straight as you can find. A good way to practice the blip technique is at times when you don't need to apply the brake (mostly because trying to practice blipping and applying the front brake at the same time is tricky). For example, while on the highway or open road you can attempt to keep your speed constant but downshift a gear or two using the blip technique, then shift back up. Your RPMs will change, but the goal is to keep your speed constant and smooth. After perfecting this, you can practice combining it with the front brake.
More information at the Sport Rider website. Note in the last paragraph that even some good racers don't use this technique. You'll have to let the clutch out smoothly and gradually if you don't blip.
Notes from the Voice of Experience
Last spring while riding at Deal's Gap there were several times that I locked up the rear tire going into very sharp turns when downshifting a little too aggressively. This causes the rear tire to skip to the outside of the turn just as you release the clutch. Nothing unmanageable, but something an expert rider probably wouldn't do.
Since then, I've been working on blipping the throttle slightly while downshifting hard through each gear. This allows the rpms to quickly raise to match the wheel speed for each gear as you downshift and eliminates rear wheel skid. It's still effectively using engine drag to slow you and also maintains the proper gear to exit the corner.
The way you do this is: While downshifting through the gears, hit the throttle hard and then let off (very quickly) just at the instant you release the clutch. You'll want to give it enough throttle to raise the rpms by about 1,500-2000. If you do this in quick succession while going through the gears (downshifting) it will become a rhythmic second-nature habit that occurs simultaneously to braking. Once this is perfected, it's possible to brake hard into a turn (before the turn) and downshift, while at the same time avoiding any rear wheel slide. This works best in the lower gears, where it's possible to quickly downshift through three or four gears much quicker than the bike slows down.