Not quite. They actually evaporate over time.
Never understood what the source of a such a gravitational field could be
The source of the field is a collapsed star or other object with mass greater than the Chandrasekhar Limit. Such an object collapses under its own mass. In General Relativity, this collapse is down to a single point, called "the singularity." We do not have a Quantum Theory of Gravity, but all bets -- and mathematical estimates based on QFT and/or the Uncertainty principle -- are that in the Quantum version, the mass cannot quite collapse to a single point.
what happens to all that energy when mass is absorbed by a “black hole”
Nothing happens to it. It's still there inside the hole. Because the hole is not a "hole," it's a region of space near the singularity. In the early 20th century, it was known these kinds of masses could theoretically exist; so massive that light could not escape. At that time they were called "frozen stars" to capture the idea that light could not escape them. Black Hole is better but still leaves room for misinterpretation. There is no hole. It looks like a "hole" to an outside observer, because things that get close enough (even light) "fall in." A better term might be Black Suckers or Black Gobblers.
OK but explain a singularity. Always thought that all objects had a “gravitational” pull that is proportional to its mass. If an object such as a star “collapses” unto itself, its mass would be a bit more dense but overall less as some of that contracting mass dissipates as energy. Then why is the gravity field so much stronger?
Shouldn't it look like a black orb? 3-D orb rather than a 2-D hole.