You can use `Normal`, documentation says it deal with special forms but `ConditionalExpression` is not explicitly mentioned there.

    p1 = y /. {First[Solve[x^2 + y^2 + x == 1, y, Reals]]} // First
>      ConditionalExpression[-Sqrt[1 - x - x^2], 1/2 (-1 - Sqrt[5]) < x < 1/2 (-1 + Sqrt[5])]

    Normal @ p1
>     -Sqrt[1 - x - x^2]