Why can't we round results of linear programming to get integer programming?
In $\mathbb R$, one can simply round down or round up to obtain an element of $\mathbb Z$. Only two choices!

However, in $\mathbb R^n$, one has $2^n$ ways of rounding to obtain an element of the integer lattice $\mathbb Z^n$. For example, if $n=100$, one has $2^{100} \approx 10^{30}$ possible ways of rounding. 

Of course, one can round all entries of an $n$-dimensional vector down or up, but do note that the Euclidean distance between the two farthest vertices of the hypercube $[0,1]^n$ is $\sqrt n$. If $n=100$, then $\sqrt{100} = 10$.

Unfortunately, to the best of my knowledge, there is no guarantee that the solution to the integer program will be found via one of the $2^n$ possible rounding schemes. 

Fortunately, some linear programs have integer solutions and, thus, no rounding is required.