The intuition behind QFT isn't the problem. I'd argue its quite intuitive: write a classical field, assume some plausible commutation relations, turn the crank. To add interactions pretend that you observe the results at infinity or whatever and take some terms of a power series representing the amplitudes, adding a cut off which you calibrate with an experiment. All fine and dandy. Just sucks that the machinery doesn't quite pass a combination of mathematical rigor and philosophical substance.