Why does the temperature of an object rise to a certain level and stop under sunlight?

It's all to do with the laws of thermal dynamics, in particular with how an object emits radiation with temperature: see the Stefan-Boltzmann law

You can see with metals, that once you heat them up they start to glow. Heat them up hotter, and they glow brighter - this shows that hotter things are brighter because as temperature increases further everything emits more radiation, so given a set power of a lamp, the jar you mention will eventually reach a thermal equilibrium, where it is taking heat equal to the amount it releases to the local environment.

You should distinguish different words:

• "Heat" - a form of energy
• "Power" - a rate of transfer of energy (an amount of heat per second)
• "Temperature" - depends on the heat, mass, and heat capacity of the object

One of the "laws" is that "heat won't pass from a cooler (low-temperature) body to a hotter (high-temperature) body". If you put the two bodies together (without insulation) then heat will pass from the hotter to the cooler. When they reach the same temperature then the two bodies are in thermal equilibrium with each other.

The situation with the lamp isn't quite like that. You might have:

• A radiant lamp
• Radiating heat into a jar
• Which is surrounded by cold air

It's not true that the jar and the air reach the same temperature; what is true is that thermal equilibrium is reached, when the rate at which heat is absorbed by the jar (from the lamp) equals the rate at which the jar loses heat (via radiation and conduction and convection) to the colder air which surrounds it.