The main problem is that you cannot achieve Libor in the markets. So the old-fashioned method of discounting at Libor doesn't work any more. As an example, if you compound up the 3m Libor with today's price on a 3x6 FRA, you won't get 6m Libor. Traditionally, that would mean arbitrage, but these days it's just a fact of life. You cannot achieve 3m Libor for the first 3m months, so the PV of that cashflow comes through a funding curve (e.g. OIS curve). At 3x6, because FRAs pay at the start discounted at the quoted rate, you can't achieve 3m Libor on that cash flow either. So the expectation that you could borrow the cashflow for the 3x6 FRA at today's 3m Libor, and lend the 6m at 6m Libor just doesn't work. You come up short, and that amount is the basis.
These days, it is safer to treat 3m and 6m Libor as two indices to which you have exposure, and which are correlated but not bound together. So doing 3m IRS gives you exposure to 3m Libor, which you can only correctly hedge or arbitrage using 3m instruments.
In USD this is simple enough, because most things are 3m. In EUR, GBP etc, longer term IRS are 6m, and short term IR is 3m, and you end up with a 3/6 basis exposure.