You should really have discussed a potential counter-offer from your current employer before accepting the new offer. Give them a deadline if necessary. "I need to accept or reject the offer from the potential new employer by Friday. Please think about whether you can and want to give me an attractive counter-offer in writing by that time."
Accepting an offer from a new employer and then deciding to go with the counter-offer is unprofessional, there is no way around it, and you won't be able to sugarcoat it very much.
If you did immediately request a counter-offer from your current employer, but they were late about making it, the unprofessionalism lies with them. In this case, you can tell the alternative employer that the counter-offer was late in coming. (Without badmouthing your current employer.)
That said, just tell the new employer as soon as possible (they will need to restart their search) that you will unfortunately need to renege on your accepting their offer, since your current employer made a more attractive counter-offer. No need to go into details. Explain that you understand and regret that this is not a good situation. As I said, you will be burning bridges somewhat.
If you are a really strong candidate, your new employer might even make a counter-counter-offer, or you might even suggest something along these lines. However, initiating negotiations like this after accepting an offer and then reneging on it might backfire badly. Tread carefully here.