if your model were of the form  `y ~ x1 + x2` This (roughly speaking) represents: 

     y = β0 + β1(x1) + β2(x2)

     Which is of course the same as 
     y = β0(1) + β1(x1) + β2(x2)

There is an implicit `+1` in the above formula.   So really, the formula above is `y ~ 1 + x1 + x2` 

We could have a very simple formula, whereby y is not dependent on any other variable.  This is the formula that you are referencing, 
`y ~ 1`  which roughly would equate to 

     y = β0(1) = β0

As @Paul points out, when you solve the simple model, you get `β0 = mean (y)`

<br><hr><br>

### Here is an example  ###

      dat <- data.frame(y= (-2):3, x=3:8)

      simpleModel <- lm(y~1, data=dat)
      simpleModel$coef
      # (Intercept) 
      #         0.5 

      mean(dat$y)
      # [1] 0.5