Why were lepers outcasts in Jesus' time?

Leviticus 13:45 And the leper in whom the plague is, his clothes shall be rent, and his head bare, and he shall put a covering upon his upper lip, and shall cry, Unclean, unclean.

Crying, "Unclean! Unclean!" tends to elicit a negative response in the hearer, causing him/her/them to stay clear of the one issuing the warning. Hence, the attitude and fear in which lepers were held in Jewish society. The people in Jesus' time were still practicing the same law that Moses received from יהוה about 1400 years before, including Leviticus 13:45.

Another answer:Lepers were kept apart from the general population and compelled to warn others away from long before the time of Jesus, for very good reason. Leprosy was a highly contagious disease, spread by both direct contact with the afflicted person and indirect contact through items touched by the leper, such as clothes and bedding.

The hygienic laws God gave the Jews regarding leprosy are recorded in Leviticus chapters 13 and 14, and go into great detail.