First of all, Rashi in Chumash says it. On the pasuk (Bemidbar 20:1) …And Miriam died there… Rashi quotes the Gemara (Moed Katan 28a) and writes: Why is the demise of Miriam written aside the parasha of the red heifer (the parah adumah)? To teach us that just as korbanos bring forgiveness, so too does the demise of the righteous.
If you want more sources I will be happy to write them, however, the Maharal in Gur Aryeh, on that Rashi writes: אמנם עיקר דבר זה מה שמיתת צדיקים מכפרת הוא ידוע
Also, the Chasam Sofer writes (in Vayikra 16:1) – כי מיתת צדיקים בעצמו מכפרת מקופיא על הציבור כשרוב עונות גורם חלילה הקב”ה נוטל הרב שבהם ר”ל לכפר על הכלל וכעין כפרת פרה אדומה
Certainly we never know exactly why Hashem does specific things, unless a Navi informs us, and we can never know if this specific tzaddik was taken away in order to be a korban or for other reasons.
It is just an expression of consolation to the mourner that possibly the deceased was a great tzaddik and was taken away in order to be a kapparah for Klal Yisroel.
The mourners who undoubtedly believe that the deceased was a worthy person, is given the feeling that he or she was even loftier than they had thought and is meant to make them feel proud and diminish their pain somewhat.