They do not charge the client, it is effectively their risk to take you on. ie. they will only do so if they think that they are likely to win, hence they only act in non-fault accidents. Basically they make their money by charging for car hire when your car is being repaired and billing it to the 3rd party's insurers. The model is fairly simple, but quite high risk for them. Put simply they get your car repaired and provide you with a replacement in the meantime, then send the bill to the 3rd party insurer. If the 3rd party refuses to pay then Europa take them to court for non-payment. In 9 out of 10 cases the 3rd party insurer would rather pay out than be dragged through the courts. In practice it seems to work well, and Europa have a very clear incentive to pursue your claim aggressively, unlike most "high street" insurers, who mostly couldn't care less.
The upside is that you should never need to pay your excess and in theory it is not a declarable insurance claim, although you should inform your insurers about the non-fault accident. In theory it should not have a negative impact on no claims bonus etc.