My understanding is that the gravitational pull of a black hole is so intense nothing can escape it, not even light. So how is it in "Interstellar" that the planets orbiting Gargantua can withstand the gravitational pull of the black hole when it appears the system's sun is being torn apart?
They ARE falling in. Just very slowly. Thanks to Inertia.
Their linear speed forward and their distance away from Gargantua allows them to reach a stable orbit around their primary gravity well. But each rotation around Gargantua shave a tiny bit of momentum as it pulls in on the planet. Their forward momentum allows it to resist being pulled in directly right now, but in a few billion to trillion years, they will get pulled into a decaying orbit, as the planets don't have a way to course correct (without external influences).
Add to that, the other planets will also exert a force on each other, and can cause stabilization issues, making this happen sooner.
The same is true for our Solar System. Influences of Planets on each other make predicting celestial body movement over long time frames hard. It's believed Mercury will eventually be pulled on too much by Jupiter causing it to collide with Venus or the Sun.
But we are talking astronomical time frames. The human race will be long dead or gone by the time this happens.
Because that's not how black holes and orbital mechanics work.
It is true that a black hole has basically so much mass (thus gravitation) that at a certain point nothing can escape its gravitational pull, not even light. That's true. But this only happens at a certain distance to the black hole, the so-called event horizon (because events beyond this can never influence events outside of the black hole).
But to the "outside world" a black hole is really just a very heavy thing, like any other planet or star, just quite a bit heavier. So planets or any other celestial bodies can just orbit around it, like the Earth orbits around the sun. Afterall you could ask the same question about why the Earth doesn't get sucked into the sun, as the sun also has a huge gravitational pull on it. But the Earth moves on a stable orbit around it (it's basically in a continuous state of freefall). In the same way the planets that orbit Gargantua move on a stable orbit around it.
What you shouldn't do is disturb that planet's orbit too much or get too near to the black hole's event horizon, then you're doomed.
I don't know why you think "it appears the systems sun is being torn apart", though. That happens at no point in the film (in fact there isn't actually a sun to tear apart there). What you might be talking about is the slightly unusual appearance of the black hole, with this glowing halo around it. This isn't a system's sun that's torn apart, but rather the black hole's accretion disc, a ring of very hot and very fast stuff that's circling around the black hole. (And it looks quite a bit weirder due to how optics work under heavy gravitational influence.)