Don Biglia's comment is pretty much spot on - the soccer team just plays somewhere else, or not at all, when the stadium is in use for the Rugby World Cup. To take a couple of examples:
- Villa Park was used for rugby matches on the 26th and 27th September. Aston Villa played away at Liverpool on the 26th September.
- City of Manchester Stadium will be used for a rugby match on 10th October. Manchester City don't play at all that weekend.
Almost every major soccer ground is a multi-use venue one way or another, so this isn't a new problem for the teams - for example, Old Trafford is hosting the Super League (rugby league) Grand Final on 10th October, which is not really any different from hosting a union match.