Why can't we solve the Halting Problem by using Artificial Intelligence?
The halting problem is not a statement about intelligence (human or artificial) it is a statement about the limits of mathematics.  It is an historically important example of an undecidable problem.

An artificial (or human) intelligence can certainly look for and find many sorts of different infinite loops in real programs.  And the halting problem doesn't say that such a thing is impossible or even very hard.

What the halting problem says is that given any program $I$ ("I" for "intelligent"), (say a really complicated AI program written in Netbeans) that looks for infinite loops we can produce a program that $I$ can't analyze.  The way we do this is a clever technique called diagonalization, which is described in the question @Raphael linked: How to show that a function is not computable?, and also in Why, really, is the Halting Problem so important?

The halting problem is similar to the Liar's Paradox in that it demonstrates the limits of mathematical and logical definitions of "truth", and "provability".  The Liar's Paradox is: suppose I say "this statement is a lie."  Is that statement true or false?  If it is true then it must be a lie, so must be false.  If it is false, then it is not a lie, so must be true.  Writing a complicated artificial intelligence (or relying on a really smart person) won't make the question about whether I lied or not any more meaningful.