The main reason is to slow oxidation, which continues at a very low rate even after lyophilization. Freezing reduces that rate further and keeps the peptide effective for years or longer. Once reconstituted, refrigeration limits any bacterial growth that might have been introduced during mixing, since room-temperature liquid would let it multiply faster. The practical approach is to freeze the dry vials for long-term storage and refrigerate the mixed ones for use within a month or so.
Peptides are presumably produced at room temperature, then lyophilized, held at ambient conditions for some unknown period, and shipped without temperature control until they reach the buyer. After that most people freeze them. Since room temperature here just means no controlled environment, is there any benefit or even a downside to freezing the powder when it will be used within a few months? Maybe freezing only makes sense if storage will last six months or longer?