Here's a decent video of a diver demonstrating how to do it. The first 2 minutes of the video they are doing something else, so skip to 2:15 mark or so.
Notice that the diver completely prepares the DSMB before blowing it up. It is unravelled, and both the buoy and the reel are in their left hand. That leaves their right hand free to handle the reg in their mouth. So they remove the reg, blow a breath into the buoy, and replace the reg. At this point, they will still be neutrally buoyant, because the gas in the buoy was a moment ago in their lungs. They will only become buoyant if they take in a big breath, but can stay neutral by taking small breaths at the "bottom" of their lungs, ie not fully inflating their lungs.
Once they do this, the buoy is halfway filled. Notice that at this point the DSMB is not attached to their body, it is loose in their hand. They do a quick check above to make sure they buoy won't become entangled on kelp or a boat or another diver or whatever. They move the DSMB and reel away from their body which will keep themselves from being entangled. And finally release some grip on the reel to allow the buoy to shoot upward. As the buoy goes up, the gas inside will expand such that it is fully inflated when it arrives at the surface (this is why DSMBs have an overinflation valve).
At that point, they can clip the line off to the reel with the double-ender, which will allow the diver to let go of the buoy and reel -- it's not going anywhere.
Some folks prefer to fill it from a drysuit or inflator hose. I guess the upside there is that they don't have to remove their reg from their mouth. The downside is that they are not transferring gas from one low-pressure compartment (lungs) to another low-pressure compartment (buoy). Instead it's from the high-pressure compartment (tank) to the low-pressure buoy. So it causes a net increase in buoyancy, versus from the lungs causes no net change in buoyancy, and makes the whole thing more time-sensitive since you'll have to swim down until the thing is deployed, which makes it harder to deploy than it would be if they could relax.