For a given group and generator $g$, the DLP is finding randomly chosen $x$ given $a=g^x$; and the DHP is finding $g^{(x\,y)}$ given $a=g^x$ and $b=g^y$, for randomly chosen $x$ and $y$.
It is known a method solving the DHP given an oracle capable of solving the DLP: find $x$ from the given $g^x$ using the oracle, then compute ${\left(g^y\right)}^x$, which is the desired $g^{(x\,y)}$. In other words, the DHP is no harder than the DLP.
It is not known any method solving the DLP given an oracle capable of solving the DHP. And thus we do not know if the DHP is easier than the DLP, or equivalent.
The citation in the question is
If the only way of solving the DHP requires the DLP, one would say that, the DHP is equivalent to the DLP
and that's clearly correct: if solving the DHP required solving the DLP (for $x$, or for $y$, or for an otherwise random instance of the DLP), then the DHP would be no easier than the DLP; and since the DHP is no harder than the DLP, the two would be equivalent.
However "the only way of solving the DHP requires the DLP" is false, or at least a gross oversimplification; making the quoted statement moot. Counter example: let $u\gets g$ and $v\gets b$; while $u\ne a$ repeat $u\gets u\,g$ and $v\gets v\,b$; output the final $v$, which is the desired $g^{(x\,y)}$.
Another observation is that the best known methods for solving the DHP involve solving a random instance of the DLP. And it there was a significantly better method, it would prove that solving the DHP is easier than the DLP. That might be a better statement of what is meant by the citation.