There are various reasons a private firm would finance a PPP project. The most common reason is that the private firm is the contractor for the infrastructure project. In order to construct the infrastructure as required by the concession agreement, the contractor needs to commit a certain amount of resources, including finance. In addition, lenders will also often looks to the commitment of (and often a guarantee from) the private firms who are contracted to conduct the concession agreement in assessing whether it would provide finance for the project.
Other private firms may have an interest in the project going ahead, to the extent that it is willing to provide certain assurances or finance. For example, a sponsor may be the operator of the project on completion of the infrastructure, a supplier of the project, or a customer (being a private firm that will take the benefit of the goods or services produced from the project).