I think they mean they train the immune system by creating an antigen, not an antibody, that mimics or is identical to the nonmutating antigen on the virus. Your immune system upregulates appropriate antibodies on its own through a feedback effect when certain cells (B cell?) that have randomly generated embedded antibodies on their cell membranes come across an antigen.
These cells begin to divide faster, and the children code for and generate the same antibody as the parent. This ups the amount of cells with that particular antibody, increasing the probability that the antigen will be targeted. They trigger inflammation to increase the flow of blood, letting more immune cells per second pass through scan the area (probably not helpful for aids, but for a cut, a splinter or a tumor that is fixed in place it is useful). They also release freestanding antibodies with the same binding profile that can disable viruses and trigger other immune cells to come slurp them up.
I think when they say they aren't taking an antibody approach they mean they aren't making an artificial antibody and just injecting it for passive immunity (sort of like injecting an immune system that can't do its own adaptation and upregulation)
The main thing with vaccines I am still confused about is what the difference between them and allergy shots is, and how do the two end up with opposite effects? I guess there is somehow a different reaction to a spike of something vs a repeated presence.
(Edit: never mind.. microspheres.. who knew?)