This is a garment that would be soaked with blood and sweat and every kind of filth, torn in spots, absolutely disgusting. Why did they even care about keeping it?
They probably didn't. It's one of many details that the writer of John's Gospel inserted into the Passion narrative so it would parallel the Messianic prophecies in the Old Testament (Psalm 22:18, "They divide my clothes among them and cast lots for my garment"), strengthening the motif of Jesus' death being the fulfillment of scripture. The question should be not why did the Roman soldiers do this, which is like asking how many children had Lady Macbeth, but rather what was the purpose of inserting this detail into the story?
Cloth of any kind was very valuable before the invention of the spinning jenny - a robe like the one Jesus was wearing would have taken hundreds of hours of labor to produce (see Spindle Whorls in Archaeology for a discussion of the immense amount of labor involved in spinning the thread necessary to make a single garment using a drop spindle). Even bloody and messed up, that fabric still had value.
But it wasn't just fabric - it was purple, a color that was produced with very rare and expensive dye made from a specific species of shellfish (murex). It was a very expensive garment.
One can imagine that the soliders probably hoped they'd manage to wash the blood off once they went home so they could flip the garment and get some easy cash. Even a torn-off piece of the garment could be a bit of a windfall.
That's the litteral reading. It's also worth considering what symbolic weight the events may have had that led the writers to include them in the story. Christ's symbol of kingship being split up among gentiles, according to throws of dice (a common way to predict the future or the will of the gods), may have been meant to foreshadow the spread of Jesus' followers out among their non-Jewish neighbors which had already begun by the time the gospels were written down. Just as dice - divination - had guided the dividing of the garments among the soldiers, messages from God sent Peter and Paul out to preech to the gentiles. Maybe John and the others wanted to set those later events up with some foreshadowing.