While braking, you should be downshifting at the same time (Brake/Down), to be in the proper gear to pull you through the corner or to pull you out of trouble. While downshifting, you need to blip the throttle, to avoid jerking the bike and/or the rear wheel loosing traction.
The physics behind this is that the current speed, current gear and engine revs must match. As you are braking for a corner, the speed of the bike goes down together with the engine revs, but this will put your revs out of the power band. To bring the revs back up while still slowing down, you must downshift.
Now, the engine drops revs whenever you roll off the throttle. Also, a lower gear at the same speed requires more revs. So, when you downshift, a blip of the throttle is needed to get the engine spinning faster and match the revs required for the lower gear.
If you don't match the revs while downshifting (or put the revs in the ball park), the rear wheel may loose traction, chatter, skid etc... unless you have a "slipper clutch" or, it's simpler cousin, the "anti-hopping clutch". These are simple devices that make downshifting safer.
The basic sequence when downshifting is:
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The idea is that if you just downshift, when releasing the clutch, the rear tire may skid or skip, since the engine has less revs that it should for that new lower gear. This is much more pronounced on dirt, where there is less traction than on tarmac.
So, proactively, you blip the throttle to get more revs before re-engaging the clutch.
What makes this hard is the fact that you are generally doing it while still pulling the front brake lever and without experience, you will jerk the bike. If that still happens after a while, make sure that your brake lever is close so you can hold one-two fingers there comfortably and not have to reach for it. Also, check the throttle play - may be you have too much and that means having to roll more than otherwise to get the revs up (which pulls the fingers and the brake lever with them).
You have to downshift while braking or while preparing to overtake someone (need more revs/power) or to prepare for an up-hill, mud, sand etc - any situation where you need more revs and a lower gear.
Quite obviously, you don't need to blip the throttle while up-shifting... the new higher gear requires less revs and the drop in revs while you roll off the throttle is enough.
The other time when you have to blip the throttle is when you pulled in the clutch to smooth out traction in a corner and you don't want the motor to loose revs: you will hear blips all the time around dirt bikes: on dirt, clutch is often use to control traction and you still need to keep the revs up, so frequent blips are in order.
For instance, you charge hard into a corner, lock up the rear wheel to slide into the corner. During this time you must pull keep the clutch pulled in, but the motor will start to drop the revs. You must then, at the same time, blip the throttle to keep the revs up... can you say 'adrenaline overload'? Yeah baby, it's why we race!
Read more Thoughts on braking.
You should also read Keith Code's excellent notes on what he calls Brake/Down - the braking and downshifting continuum.