The Roman numerals often indicate that the notes should be played on a particular string. For the Violin,
I = E-string, II = A-string, III = D-string, and IV = G-string. In your example, the first and second violins are expected to play the passage on the G-string in 6th position (nearly halfway up the string). The violas are also meant to play on the G-string (III for viola). Notes played high on the G-string have a very dark and beautiful quality which cannot be reproduced on the D-string.
There are also instances where Roman numerals indicate the position, rather than the string, i.e.
I = 1st position, III = 3rd position, V = 5th position, VII = 7th position, etc. Caprice #6 by Pierre Rode for example uses Roman numerals for such position indications.
The violinist usually can figure out which is meant by context.
Examples of Roman numerals for string indications
Paganini Caprice 18 - IMSLP Link
Opening Corrente section played entirely on the G-string up to a high E reached by stretching from 7th position.
Bach Partita 3, Prelude -
Measures 64-66 are played on an open A and fingered D-string.
Measures starting at 67 are played on an open A, with fingered D and G-strings.
Sarasate Zigeunerweisen -
Example of Roman Numerals for Position
- Caprice #6 by Pierre Rode