What does the scientist mean when he says that white blood cells would not be “fast enough or smart enough if we hadn’t whipped them along by a prior immunization”?

To rephrase your question, I think you're asking "Why does immunization increase the speed and effectiveness of clearing an infection?"
Immunization is the process by which exposure to a small amount of pathogen (virus or bacteria) protects from future infection. To immunize against a particular infection, you are given a vaccine. The following critical steps happen when you receive a vaccine and are successfully immunized.
1. B-Cells (a type of white blood cell) start creating antibodies to the infection. The first time you encounter a pathogen, the type of antibody you create (IgM) is less effective at triggering an immune response than the antibody you create every time after (IgG). Additionally, this switch from low- to high-effectiveness antibodies can take anywhere from days up to a week. With immunization, the body "remembers" the IgG it made before and can mount a better immune response faster (see step 4).
2. In some cases, your body also activates T cells (a type of white blood cell), which are responsible for clearing intracellular infections like viruses.
3. The activated T cells or antibody-producing B cells expand, and as they expand, their DNA replication is slightly error-prone. This error-prone replication helps the white blood cells select for better and better binding to the immunized pathogen.
4. After the small amount of pathogen delivered in the vaccine is cleared, your body creates special types of white cells known as "memory B cells" and "memory T cells." These cells are how your body remembers it was previously infected with that particular pathogen. When you encounter a pathogen a second time, these memory cells help activate the immune system faster and stronger than before.
In summary, immunization helps your immune system respond faster to a second infection through 1) Helping create effective antibodies and T cells to the immunized pathogen and 2) forming memory T and B cells, which quickly reactivate the immune system.
I've included a link to a diagram detailing this process below.
https://www.newhealthadvisor.org/Primary-Immune-Response.html