One way to look at it is that gravity warps the “fabric” of space. A massive object in empty space makes “straight” lines bend toward it. And the more massive the object, the greater the warping and the more those lines bend.
And just like a ball rolled or kicked along an uneven surface, the faster you go, the less this warping affects you.
Light moves really fast, so the effect is relatively small. But black holes’ mass is enormous, so it bends those straight lines enough to capture light that comes too close.
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