Why only bride needs to move to groom's place after marriage and not groom to bride's place?
Because in Vedic marriage, the daughter belongs to the groom's family after marriage, and she is no longer part of her birth parents' family. This is also why her gotra changes to her husband's gotra, and also why today her last name is changed to her husband's last name.
Before the daughter is married to a man, she is "given" as a dAna (donation) to a suitable bridegroom:
Manu 9.88 - One shall give his daughter in the proper form, even though she may not have attained (the age), to a bridegroom who is of exceptionally distinguished appearance, and her equal.
This "giving" means transfer of family membership and thus responsibility of the girl to the groom's family:
Manu 5.146 - In childhood she should remain under the control of her father, in youth under that of her husband, and on the husband’s death under that of her sons; the woman should never have recourse to independence.
From that point on, the wife serves the husband and father and mother in laws, and all the Pitr offerings the couple make go to her new family's ancestors.
That is why this practice is there.