The results seem impressive enpugh to recommend the paper without knowing how they are obtained. (I decided to do so rather than postpone the recommendation till a time in which I will be able to know more about the technical ideas.)
Added (a few days later): Reading Sec 1.4, it seems that the results are obtained by a careful application of the "testing by implicit sampling" method of Diakonikolas, Lee, Matulef, Onak, Rubinfeld, Servedio, and Wan (see Chapter 6 in my introduction to property testing book). In particular, an ingredient I don't recall being used before is running a learning algorithm (also via implicit sampling) to obtain a candidate function supposedly closed to the tested function when restricted to the relevant variables.
We give improved and almost optimal testers for several classes of Boolean functions on $n$ inputs that have concise representation in the uniform and distribution-free model. Classes, such as $k$-Junta, $k$-Linear Function, $s$-Term DNF, $s$-Term Monotone DNF, $r$-DNF, Decision List, $r$-Decision List, size-$s$ Decision Tree, size-$s$ Boolean Formula, size-$s$ Branching Program, $s$-Sparse Polynomial over the binary field and functions with Fourier Degree at most $d$.
See ECCC TR19-156.