If the speed of light is always constant then light should escape from a black hole because if directed radially outwards it only needs to travel a finite distance to escape, and at a speed of $c$ it will do this in a finite time.
Since light cannot escape from the black hole that means the speed of light must be less than $c$ near the black hole. How can a black hole reduce light's speed?
Best Answer
Have a look at the question Speed of light in a gravitational field? as this shows you in detail how to calculate the speed of light in a gravitational field.
I haven't flagged this as a duplicate because I'd guess you're not so interested in the details but rather how the speed of light can change at all. You've probably heard that the speed of light is a constant, so it's a fair question to ask why it isn't constant near a black hole. The answer turns out to be quite subtle.
In special relativity the speed of light is a global constant in that all observers everywhere will measure the same value of $c$. In general relativity this is still true only if spacetime is flat. If spacetime is curved then all observers everywhere will measure the same value of $c$ if the measurement is done locally.
This means that if I measure the speed of light at my location I will always get the value $c$, and this is true whether I'm sitting still, riding around on a rocket, falling into a black hole or whatever. But if I measure the speed of light at some point that is distant from me I will generally get a value different from $c$. Specifically if I'm sitting well away from a black hole and I measure the speed of light near it's surface I will get a value less than $c$.
So the answer to your question is that in GR the speed light travels isn't always $c$.