M_p bound: For a second order system, the equation M_p = exp(-pi*zeta/sqrt(1-zeta^2)) gives the relationship between the damping ratio zeta and the overshoot M_p. Given a desired M_p, this equation can be solved for the corresponding zeta (this is shown in the table on slide 9). If you want M_p to be larger than a certain value, then you need zeta to be less than the computed value.
Given the bound on zeta, the next question is how to get the constraint on the pole location. Notice that the poles are at s = -a + bj, where a = zeta omega_n and b = +/- omega_d. If zeta is small, then omega_d is approximately omega_n and hence we can say that -a/b ~= zeta. Thus, the ratio of the real to imaginary part of the pole must lie to the *left* of the line given by a * zeta = -b (this is the line above the x axis; a similar argument holds for the mirror image).
A similar argument can be used to generate the constraint corresponding to the settling time, T_s.