How does the entropy in the universe changed over time?
All real processes - have the effect of increasing the entropy of the universe. That is the second law of thermodynamics.
The Sun, and every other star, is radiating heat into the universe. But they can’t do it forever. Eventually the heat will have spread out so much that there won’t be warmer objects and cooler objects. Everything will be the same temperature. The same, very cold, temperature. The vast majority of the universe is already screaming cold, so the heat death of the universe is just about burning what fuel there is and mixing the heat so created into the ever-expansive, cold, and unyielding cosmos. Both the burning of fuel (mostly through fusion in stars) and the distribution of heat are processes which increase entropy.
Once everything is the same temperature, the universe attains a “steady state”. With no energy differences, there’s no reason for anything to change what it’s doing (all forces can be expressed as an energy imbalance or “potential gradient“). Heat death is the point at which the universe has finally settled down completely (or almost completely), and nothing interesting ever happens again.