The Laplace transform resolves a function into its moments.

The Laplace transform is related to the Fourier transform, but whereas the Fourier transform expresses a function or signal as a series of modes of vibration (frequencies), the Laplace transform resolves a function into its moments

Can someone explain what does "the Laplace transform resolves a function into its moments" mean?

I disagree with the statement with regard to the $FT$, but with regard to the Laplace transform, I believe what is meant is that for a function $f(t)$ the Laplace transform $$ F(\lambda)=\int_0^\infty f(t) e^{-\lambda t} dt $$ can be expanded as $$ F(\lambda)=\sum {1\over n!}(-\lambda) ^n \int_0^\infty t^n f(t) dt $$ (when permitted) which can be iterpreted as $$ F(\lambda)=\sum {(-\lambda)^n\over n!}\langle t^n\rangle $$ where $\langle t^n\rangle$ is the $n^{th}$ moment of $f$, defined as $\langle t^n\rangle =\int_0^{\infty} t^n f(t)dt$.

The same interpretation of the Fourier transform does not "express[es] a function or signal as a series of modes of vibration", rather it is a very similar (but two sided) moment expansion. The mode of vibration decomposition is if you consider each $\omega$ in the $e^{i\omega t}$ as a frequency, not by expanding in $\omega$.