If you look at some simple progressions in the key of C major, it is easy to see how the system of Roman numerals works. The following example contains the chords C, F and G. Since these chords correspond to the first, fourth and fifth degrees of the C major scale, the progression could be described as I IV V I in the key of C.
This one contains the chords C, Em, Dm and G which correspond to the first, third, second and fifth degrees of the C major scale: therefore the progression could be described as III II V in the key of C.
Even when you see a more complex piano part, it is usually based on a simple underlying chord progression. Try reducing the following examples to block chords, then try taking any progression and making a piano part out of it by using broken chords, arpeggios, adding notes to the chords and varying the rhythm.
This part looks more complex and is harder to play, but it is based on an even simpler underlying chord progression. Analyze the progression underlying everything you learn and experiment with different ways of playing it.