y ~ x
Gives significant positive relationship between x and y
y ~ x + (group)
Gives significant inverse - negative - relationship between x and y. This seems to be due to the fact that small groups had different y and x associations than large groups.
I have to use the second model in my article. How should I discuss such statistical phenomena in my article?
May I say, that y and x relationship was different between small and large groups, leading to inverse results...
How precisely you should report your model depends on your context - we can't help you here.
The size of the groups is usually not the culprit, it's simply the different relationship between the predictor and the response for different groups, specifically an intercept that differs between groups. If you describe this, I would say you are doing the right thing.
Consider showing a plot of $x$ against $y$, with different groups in different colors.
What you have is an example of Simpson's paradox. I would recommend that you mention this, as it will allow readers to understand the problem. We also have a simpsons-paradox tag.
