A filter is a mathematical operator that serves to convert an original time-series into another time-series or function. The purpose is to remove some particular features (e.g. trends, business cycle, seasonalities and noise) that are associated with specific frequency components.
To get the basic idea think of a moving average which is nothing but a filter to remove (supposed) noise.
In the case of the fourier transform the time series is transformed from the time domain into the frequency domain.
Mathematically the filter is applied in both cases by convolution of the original series with a coefficient vector (basically nothing but the dot product). In case of the moving average the coefficient vector is a rectangular function, in case of the fourier transform you basically use some kind of trigonometric functions (via the complex exponential function) to extract the frequencies.
You can find more here:
Analysis of Financial Time-Series Using Fourier and Wavelet Methods by Philippe Masset
To get a general intuition about the fourier transform you can find many excellent answers here: