Sure thing. This kind of notation comes directly from the sheet music. The roman numeral names the chord relative to the key (1 chord, 2 chord, etc.) and it's quality (major, minor, etc.) and also tells how many notes are in the chord. The numbers beside the roman numerals tells the intervals of the other notes of the chord from the bottom note. I'll use the chord C / C-E-G as a visual example:
Root Position - I (it can also be written I5/3, I5, or I3)
This notation is for 3 note-chords. This means that the root is the lowest note (the note on the very bottom of the staff), and the other notes are a third and 5th above that (C-E = 3rd, C-G = 5th). So, using just note names it would look something like this:
G
E
C
C
The reason why only the I can be written for the root position chord is because it will never be confused with an other chord symbol.
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1st inversion - I6 (it can also be written I6/3)
This notation is also for 3 note-chords. This means that the 3rd is the lowest note, and the other notes are a third and sixth above that (E-G = 3rd, E-C = 6th).
G
E
C
E
This symbol can be written as I6 for the same reason as the root position chord can be written with just an I, because it can't be confused with any other chord symbol.
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2nd inversion - I6/4 (I think this one can be written I4, but I'm not 100% sure on it)
This notation is also for 3 note-chords. This means that the 5th is the lowest note, and the other notes are a fourth and sixth above that (G-C = 4th, G-E = 6th).
G
E
C
G
This chord symbol has to be written 16/4 because the previous inversion has a I6 as well. This distinguishes it from the 1st inversion chord.
There's probably a little more to it than this, but this is the basic idea.