It started with the creation when God rested on the seventh day from his labors. It was further taught in Old Testament times with the Ten Commandments given to Moses. The commandment given to Moses said: 
10 But the seventh day is the sabbath of the Lord thy God: in it thou shalt not do any work, thou, nor thy son, nor thy daughter, thy manservant, nor thy maidservant, nor thy cattle, nor thy stranger that is within thy gates: 
11 For in six days the Lord made heaven and earth, the sea, and all that in them is, and rested the seventh day: wherefore the Lord blessed the sabbath day, and hallowed it. 
 
Similarly to Old Testament times where they were commanded not to have any one work on the sabbath - in our day it carries over to avoiding activities that put people in a position where they have to work. So for example, when we go out to a restaurant on the sabbath we are participating in having them work on Sundays. So we are counseled not to engage in those types of activities. 
 
Here is a good overview of Sabbath: Link 
 
Here are some additional references - including talks - about it: Link

Why do Mormons keep Sunday instead of the Seventh day Sabbath found and commanded in scripture?

God rested on the seventh day after creating the earth. That day was sunday.