Sorry if this is too basic, but I'll try to describe what the Fourier transform means (also, warning: I'm not good at short explanations). Any "time series" (signal) can be described as the sum of individual sine waves, each with a different frequency. Any one frequency will have a certain amount of power that is not necessarily the same as other frequencies' power. Where your notes talk about "patterns" in the data, that's what I mean by these individual sine waves that we sum up. The important point that the Fourier transform is based on is that any signal can be approximated by a sum of sine waves of different frequencies (the classic example is the square wave). Even if the signal is aperiodic, you can construct a set of sine waves of different frequencies and different powers that approximate your signal. As you use a larger number of sine waves, you get a better approximation.
With sound, power corresponds to loudness. If you play two notes on the piano at the same time, one loudly and one quietly, then one note "has more power in the signal" (is louder) than the other. The Fourier coefficients are essentially the powers of each frequency. Say you have a simple signal that is the sum of 3 sine waves: one at 20Hz, one at 31Hz, and one at 49Hz. If you calculate Fourier coefficients of the signal between 1Hz and 100Hz and plot them, that will be something like your "power spectral density" - how powerful/present each frequency is in the signal. This plot would be zero everywhere except for peaks at 20, 31, and 49 Hz, because all frequencies except those three have zero signal power.
This can be useful for filtering, because sometimes the signal and noise exist in nonoverlapping frequency bands (let's say your signal is only the 31Hz component; then low frequency noise is 20Hz and high frequency noise is 49Hz). You can look at what they are with the PSD/Fourier coefficients plot, and that helps you decide where to set your cutoff frequencies to filter out unwanted frequency bands (aka noise).
In practice, all of these values will be ranges instead of individual values. So, low frequency noise might be all frequencies <20Hz, high frequency noise might be all frequencies >100Hz, and the signal might exist in the 30-80Hz band. You can then filter appropriately (high-pass filter for frequencies <30Hz, low-pass for frequencies >80Hz). It's not always clean like this: sometimes the different bands overlap (e.g. if signal has 50Hz-500Hz components and noise has frequencies >300Hz, then you have interference between 300 and 500 Hz that is harder to separate out). To make these claims you need to know some characteristics of the signal, which you often do. For example, just above middle C is the note "A" at 440 Hz, and my speaking voice is usually more than an octave below that. If you want to remove higher frequency noises (say, a dog is whining in the background that you want to suppress), you can low-pass filter at 220 Hz (1 octave lower = 1/2 frequency) and know that my voice will still be there.
There are a few ways to filter in Matlab. If you search for band-pass filters you can find something that's easy to implement (aka matlab functions that band-pass filter). You can also search for low-pass and high-pass individually; depending on the filter it doesn't make a difference if you do a band-pass vs. if you do the two steps separately.
Hope that helps, good luck!