The explanation for OEIS:A000335 states:
Euler transform of A000292
where A000292 is the tetrahedral numbers.
According to MathWorld:
There are (at least) three types of Euler transforms (or transformations).
I am not really experienced enough with math to understand any of the three transformations listed, so my question becomes: which Euler transform is used in A000335, and what does it mean?
The answer is based on generating functions. The Euler transform of the sequence $(a_1,a_2,a_3,\dots)$ is the sequence $(b_1,b_2,b_3,\dots)$ whose generating function is $$B(x) = 1+b_1x+b_2x^2+b_3x^3+\dots = (1-x)^{-a_1}(1-x^2)^{-a_2}(1-x^3)^{-a_3}\cdots$$ and the name comes from Euler's work on the partition numbers sequence which is the Euler transform of the all $1$s sequence. Note that by definition $b_0=1$ and this is sometimes included.
In this particular case of Euler transform of A000292 we calculate $$ 1+1x+5x^2+15x^3+45x^4+\dots=(1-x)^{-1}(1-x^2)^{-4}(1-x^3)^{-10}(1-x^4)^{-20}\dots$$ and note that the offset for A000335 is $1$ and thus $b_0=1$ is not in the transformed sequence.
Read the official information on transforms in OEIS for which there is a link in A000335 itself.
In the MathWorld article, the third definition is the one used in OEIS and there is a reference to the 1995 EIS book by Sloane and Plouffe.