Why is reaching escape velocity the only way to leave a planet or star?

If you flew at 1/100th the escape velocity in a vertical direction, you would eventually escape the gravity, yes, but you would be under powered flight constantly and need more fuel than is probably available on the earth by the time you left the gravity well.

Escape velocity means you get to that speed, turn off the thrust and you're free.

By your logic, you could never escape a black hole since the escape velocity exceeds the speed of light.... no cigar on that one.

Wrong it will not work the longer u fight the gravity the more fuel u will use. Trying to escape a black hole that is a joke. It is like climbing a Lauder will not work.

There is more to a black hole than merely its escape velocity, although it's interesting that a Newtonian calculation of the escape velocity accurately predicts the location of the Event Horizon.

In GR, however, spacetime itself is curved by gravity. This means that starting from any point within the event horizon, all possible trajectories lead eventually to the singularity, regardless of velocity.

If you had the means of achieving thrust for an indefinite period of time, with thrust just equal to the pull of gravity plus a small delta, the sum total of all this excess thrust would be equal to escape velocity by the time you get out of the gravity well.
Escape velocity is defined as the speed you need to have where you are to be able to make it out of the gravity well with zero speed left, i.e. zero speed at infinity.

Now, if you want to pull out of a black hole gravitational pull, and that you are already beyond the event's horizon, the acceleration you need just to stay there is infinite, you need your small increment of velocity to be higher than the speed of light.  Note that if you are far enough from a black hole, outside of the event's horizon, then the same rule that applies to a planet or a star is valid: you CAN get away, if you have enough thrust.

You never get beyond a planet's (or star's) gravity.  Gravity goes to infinity.  It does, however, diminish as the square of the distance increases.

The escape speed is directly related to the gravity.  As you move away from a planet (or star), the escape speed is less.  At Earth's surface, the escape speed is 11.2 km/s.

As you move away from Earth's surface, the escape speed is less and less (never 0 in theory but, in reality, you can get far enough that some other planet's (or star's) field becomes stronger than Earth's).

Therefore, by keeping a constant speed of 0.01 * Escape-at-Surface, you will eventually reach the escape speed for the distance you are at.  

Then you will have escaped.

Escape velocity is actually a rather simple concept.  Either an object has enough kinetic energy (velocity) to be converted into potential energy (elevation) to escape a planet or it will fall back to earth.  Throw a baseball up in the air and it will return to earth because it was not initially traveling at about seven miles per second (neglecting wind resistance).  As the baseball rises it's kinetic energy is converted to potential energy as it slows down (and no energy is lost!).  Of course, a rocket with a nearly inexhaustible supply of energy could add enough energy to initial velocity to escape.  A black hole is an entirely different animal!  Within a black hole (time horizon) space is so curved that if a photon is thrown up (by a flashlight?) it will have too little initial velocity (the speed of light) to escape and will return toward the center of the black hole where all matter is condensed into a singularity so dense that it could occupy the period at the end of this sentence(.)  The gravitational attraction of such concentrated mass is great and approachable, whereas we are limited to approaching the earth's center of gravity by the earth's surface (not to mention the heat with depth).  If the earth's mass could be condensed into a singularity it too would be a black hole (and invisible in space).