I am familiar with LP relaxation for ILP (or IP). Assume we concern with integer minimization problem, which we formalize using ILP; we then relax the ILP into LP and we say that the LP provides a lower bound for the ILP. Why this is correct?

I do understand that a feasible solution for the ILP is a feasible solution to the LP, and the reversed is not always so, i.e. a feasible solution for the LP is not necessarily a feasible solution for the ILP.

Can one please point out and explain briefly why it is so?