From an electrical engineer, here’s what this means as an ELI10.
The “Thomson effect” is a phenomenon where running electrical current through a material changes how heat moves through that material. For example, if you have a wire and heat the middle of it over a flame, when you apply an electric current you notice the wires temperature change in different places. Different materials/metals behave differently. Some (like Iron) will appear to “push heat toward the cooler parts” when a current is run through them while others (like copper) will appear to “pull heat toward the hotter parts” when a current is run through them. It’s an interesting interaction between heat, electricity, and certain materials.
The “Transverse” Thomson Effect is a theoretical (until now) phenomenon where a magnetic field is added to the equation. The theory was that the addition of a magnetic field could move the temperature changes to the sides of the electrical current, instead of simply forward/backward along the current. So if you heat a plate instead of a wire, you would see temperatures change on the left/right instead forward/backward. This has been very difficult to test accurately because of the complexity in how all these things (heat, material, electrical current, and magnetic field) can interact.