I will try to answer this but keep in mind that I am not an expert; I am just some guy on the internet. Fixed effects are factors in the data whose level you control and limit to certain values. Random effects are factors that contribute to the outcome but whose levels are not fully sampled or even, perhaps, understood. For example, in a medical study you might be measuring the concentration some blood component and you have a fixed effect with two levels:
- treat with new drug
- do not treat with new drug
A random effect might be the identity of the hospital where each patient is treated. It is not practical to sample all possible hospitals and you might not even know just why different hospitals give different results. The label of "hospital" might be a stand in for different funding, population, medical culture or many other things.
When you look at the raw results, not taking hospitals into account, you might find a very wide spread in the concentration in both levels of the fixed effect and you cannot see any effect above the data variance. Adding the "random effect" of the hospital, you might find that patients at different hospitals start with different blood concentrations and that there is a clear difference between the two fixed effect levels within each hospital. The random effect of the hospital accounts for much of the observed raw variance and accounting for that allows you to see the fixed effect.
Does that help?