In a Carnot engine the net entropy changein a cycle is zero. But in an irreversible engine operating between two temperatures the net entropy change in a cycle is positive. As I have understood, this means the irreversible engine tends to lose more heat at lower temperature than the Carnot engine. Why is it so?
Net entropy change means entropy change of the world (world means system plus environment). Carnot cycle is a reversible cycle. For a reversible cycle, world entropy change is zero. Because both of system and environment return to their initial states when cycle is completed. But, for an irreversible cycle, world entropy change (net entropy change) isn't equal to zero. Although the system returns to its initial state, but the environment doesn't. So, net entropy change won't be zero and according to Increase in Entropy Principle, it will be positive.
If the engine is operating in a cycle, the change in entropy must be zero, since entropy is a function of state. However, the heat taken in from the hot reservoir divided by the temperature of the hot reservoir must be less than the heat released to the cold reservoir divided by the temperature of the cold reservoir. This means that the net entropy entering the system during each cycle is negative, and equal in magnitude to the entropy generated within the system during each cycle. So the net change in entropy over a cycle is zero.
In any process total entropy can not decrease. It's s second law of thermodynamics. But still why is it so? I can not give a good explanation. Any system consists of atoms, photons, etc., these particles move around, collide. Motion of each of them is described by laws of mechanics. It turns out that laws of mechanics have a very interesting property: thermodynamics laws can be deduced from them. It's very interesting because at the first glance it looks like they do not have anything in common!
So, the entropy can not decrease. It can only increase or remain the same. That immediately means that at reversible process entropy remains the same. (if it increases we could reverse the process and decrease it).
If during some process entropy increases this process must be irreversible. Otherwise we would reverse it and decrease the entropy.
"Reversible" and "entropy remains the same" means almost the same. Almost. There remains a possibility that there could be some process such that the entropy during this process remains constant, but it's still irreversible. I can not think of an example of such a process.
In you question you mention some irreversible engine and ask, why the net entropy increases. Hmm. That depends on type of engine!
In ideal Carnot engine gas contacts heater and gets energy from it when both gas and heater have same temperature. But the less is temperature difference the slower is the process of transferring the energy. In not ideal Carnot engine the gas is slightly cooler than the heater. Heat is comes from heater to a slightly cooler gas - and here the entropy increases. Same situation is when gas is cooled by cooler: temperature of gas must be higher that the temperature of cooler, heat transfers from a gas to cooler ans entropy increases.