What exactly is the accretion disk? Is it really possible for material such as dust, gas and other stellar debris that come near the black hole, doesn't fall into it, instead forms a flattened band of spinning material?
A basic answer:

To us on Earth, it looks like everything falls mostly straight down and hitting the Earth.   We almost never see something falling, trapped in Earth's gravity and missing the Earth but in space that happens a LOT.

If we lived in space our observation would be very different.   Most of the stuff we see flies around, in orbits, around the Sun or Earth without colliding into them.  The reason for this is tangential velocity.  It's actually hard for something to hit something else in space, even with gravity, because space is (shockingly) mostly empty space.   

Planets, the Sun, black holes, etc, are very small relative to their gravitationally dominant regions or hill spheres.  

Why is it hard to hit the sun?  (this applies to black holes too, even more so)

Why is the solar system flat?

Mostly, objects caught in the gravitationally dominant region around a black hole will just orbit the black hole.   

When an object passes inside a black hole's Roche limit, it will begin to break apart into smaller parts, and those smaller parts into smaller parts, and close enough, pretty much all matter just becomes a bunch of debris, kind of similar to Saturn's rings but much faster moving and more dense.   Ring systems can form inside the Roche limit around any massive body, large moon, planet, sun or star, though stars which emit lots of light and heat aren't good for the stability of ring systems.  

Ring systems, kind of like Saturn's, can form around black holes when there is a collision or when an object passes too close and ring systems tend to flatten out over time due to collisions.  The angular momentum remains constant but the up or down motion of objects cancels out.  The flat accretion disk is actually conservation of momentum, it forms naturally, even if it seems like a lot of wasted space, the matter doesn't know where else to go but into a flattened, quickly rotating disk.  

Because of the laws of relativity and gravitational waves, inside about 3 Schwarzschild radii, the ring material much more quickly spirals in towards the black hole, moving faster into a smaller region, getting very hot and extremely magnetic and that's generally what's called the accretion disk.  (Ring systems are generally further out and not hot or spiraling in, though they are loosely similar)  

All happens outside the black hole.   Nothing that happens inside the black hole gets out, but what happens outside the black hole in the accretion disk can be surprisingly bright and energetic.  Brighter than the brightest stars.

It might seem like stuff should just fall into a black hole like a vacuum sucking things in, but in gravity in space doesn't work like that.  Anything with zero tangential velocity will fall straight in (kinda-sorta, there's also the dragging space around it problem for rotating black holes), but for simple non rotating black holes, Only an object with close to zero tangential velocity will fall straight, but almost nothing that moves through space has zero tangential velocity relative to it's nearest large gravitational body.  This applies to dust, gas, comets, meteors, planets, stars, anything that passes close to or orbits a black hole.

That's the overall gist of it.  Corrections welcome.