I understand that the spike protein is the most saturated target of the known immunological study, vaccines have been mainly targeting this protein. Normal immunity works on the current variant of infection and will react to these mutations.
As an aside, the protein is very interesting, check out this[0] linked animation of the viral insertion, note the dramatic conformal changes.
The mutations in particular variants might achieve a little bit of infection speed, and a little immunological camouflage, both seem to support allowing faster replication.
I don't know because I am not a doctor or virologist, but I would conjecture that the vaccine still prepares our immune responses to target this kind of protein, so it can somewhat more easily learn the new signature and produce antibodies.
[0] length: 01:54 title: Animation of SARS-CoV-2 entry into human host-cell - https://www.youtube.com/watch?v=Xuc9D4LVJdg