Local currency collection is a fuller name for local collection, the practice by which a business receives payments from customers abroad through domestic accounts and payment systems in the customer's own country, instead of asking them to send an international wire. The added words stress the currency side: the payer pays in the currency they earn and hold, and conversion happens afterwards, on the business's side.
A provider typically gives the business account details in each market, often in the form of a virtual account, so that a buyer in Britain pays in pounds by a domestic transfer and a buyer in the euro area pays in euros. The provider receives the funds under a collection on behalf of arrangement, then either holds them in that currency or converts them into the business's chosen settlement currency and pays them out.
For the payer this removes wire fees and uncertainty over the exchange rate. For the business it moves the conversion decision, and the FX markup, under its own control, and lets it accept whichever local payment method customers prefer.