There are several suggested reasons for this custom, most perfectly rational and practical, while some more Kabbalistic. Here are the answers I have seen:
(1) Mirrors can be a cause of joy – of admiring oneself or becoming preoccupied with his external appearance. It is inappropriate to focus on such when one is mourning or in the presence of mourners.
(2) Prayers are normally held in a house of mourning, and we may not pray in front of a mirror. This is both because of the distraction and because it looks like the person is praying to himself (Mishna Berurah 90:71).
(3) There was a custom in Talmudic times to overturn the beds in a house of mourning. One reason given for this was to avoid the temptation for marital relations during the week of shiva, which are forbidden. This is not in practice today, but possibly as an alternative, the mirrors are covered to lessen the desire for relations (based on Chatam Sofer).
(4) According to the Zohar, the images of people which appear in mirrors could have spirits emanate from them. At such an inauspicious time as when suffering the loss of a loved one, one may be especially vulnerable to the potential harm caused by such spirits. Thus, the mirrors are covered (Ya’arot D’vash).
(5) Evil spirits reside in a house of mourning: they fill the void left by the loss of a human life. Although we cannot see them, their reflection may appear in a mirror. Thus, the mirrors are covered (Ginzei Yosef p. 320).
(The first 3 reasons appear in Kol Bo Al Aveilus p. 262 and are cited in Mourning in Halacha by ArtScroll Publications (p. 179 footnote 8). The fourth appears in Ta’amei Minhagim, p. 435.)
See: Covering Mirrors in House of Mourning, July 21, 2017 | by Rabbi Dovid Rosenfeld
וַתִּתְפְּשֵׂהוּ בְּבִגְדוֹ לֵאמֹר וְגוֹ׳ בְּאוֹתָהּ שָׁעָה בָּאתָה דְּיוֹקְנוֹ שֶׁל אָבִיו וְנִרְאֲתָה לוֹ בַּחַלּוֹן אָמַר לוֹ יוֹסֵף עֲתִידִין אַחֶיךָ שֶׁיִּכָּתְבוּ עַל אַבְנֵי אֵפוֹד וְאַתָּה בֵּינֵיהֶם רְצוֹנְךָ שֶׁיִּמָּחֶה שִׁמְךָ מִבֵּינֵיהֶם וְתִקָּרֵא רוֹעֶה זוֹנוֹת דִּכְתִיב וְרֹעֶה זוֹנוֹת יְאַבֶּד הוֹן
This special religious relationship between father and son is most poignantly expressed by a famous Talmudic Commentary on a critical moment in the life of Joseph in Egypt, described in this week’s Biblical reading. The young and handsome Joseph, having been sold into Egyptian slavery by his jealous brothers, is purchased by Potiphar,the Egyptian Minister of Culinary Arts, who quickly appoints the Hebrew his steward, in charge of all internal and household affairs. The minister’s wife, obviously attracted by Joseph’s ability and charm, attempts to seduce him. “And (Joseph) refused” (Gen. 39:8), cries out the Biblical text – but with the drawn out and multi-trilled cantillation known as the shalshelet. This is explained by Rabbi Samson Raphael Hirsch in his nineteenth century Biblical commentary to imply that Joseph took a long time in refusing, that it was difficult for him – a stranger in a strange land - to resist the advances of such a beautiful and powerful woman. What gave him the inner strength to resist? “The persona of his father (Jacob) appeared to him in his mind’s eye”, suggest our Talmudic Sages (Rashi on Genesis 39:11, citing B.T. Sotah 3).
Rav Haim Sabbato, well-known Talmud teacher and author, recounts that once, when lecturing to a non-religious kibbutz, he mentioned this incident regarding Joseph, and the response was cynical disbelief. At such an intense, erotic moment, the very last image in Joseph’s mind would be his aged father, his audience insisted. Rav Sabbato suggested to them what I believe was an ingenious interpretation. In Biblical times, only the very rich had mirrors, and then only in the bed-room. Hence Joseph had never seen how he himself actually appeared. When ushered into Mrs. Potiphar’s boudoir, he saw his image for the first time in the mirror hanging on her wall – and Joseph was the exact physical replica of his father Jacob (Rashi on Genesis 37:3). Now Joseph did know how his father looked – and at this point of his life and suffering, he most probably had a beard which was turning grey if not white. In other words, Joseph thought he was actually seeing his father Jacob in the mirror of Mrs. Potiphar’s boudoir; and he immediately sensed hearing his father’s teachings of morality and ethics. Because of this he found the moral strength to resist temptation.
Rabbi Shlomo Riskin on Parshat VaYeshev
During the seven days of Shiva we are discouraged to see ourselves in a mirror.... we are to think only of the deceased and our loss (שֶׁמֵּתוֹ מוּטָּל לְפָנָיו). After the shiva, we remove the mirror and the mourner can see the face of the deceased in their own face.... as an inspiration for a life well lived.