Although you will be using their system for some very sensitive banking information, Trustly is authorized and supervised by the Swedish Financial Supervisory Authority (Finansinspektionen), making it that much safer and reliable.
Currently owned by Nordic Capital, the company also holds a European Payment Services Provider (PSP) license in accordance with the Payment Services Directive (PSD, 2007/64/EC). Besides, it has TÜV certification for data protection in Germany.
Overall, Trustly is known to use the highest encryption standard available, on top of the banks' security system.