On Islamqa.info they discuss different opinions of a particular topic from the 4 schools and perhaps others then say "... But the strongest opinion is..." The same applies on Q & A shows on channels such as Huda TV or Islam Channel. How do the scholars determine the strongest or most correct opinion?
Logic or reasoning.
Basically, no-madhhab sects consider that it is an individual's responsibility if he has studied the basics of Islamic Law (which are mostly the same across all maddhabs) to himself decide what he believes to be the more supported opinion by evaluating the evidences of the opinions with his own reasoning.
If he has not studied Islamic Law, it is considered best for him to find a scholar he knows personally or trusts and follow his opinions on these matters.
This is explained in an IslamQA answer:
What should the Muslim’s attitude be towards differences that arise between the scholars? In other words, which scholarly opinion should the Muslim follow in matters where they differed? The answer depends:
1 – If the Muslim is one who has studied shar’i knowledge and learned its basic principles and minor issues, and he can distinguish right from wrong with regard to scholarly views, then he has to follow that which he thinks is correct and ignore that which he thinks is wrong.
2 – If he is one of the rank and file, or has not studied shar’i knowledge, and thus cannot distinguish between right and wrong with regard to scholarly views, then he must follow the fatwa of a scholar whose knowledge he trusts and who he believes to be trustworthy and religiously committed, whether he is from his own country or another country, and differences between scholars will not matter after that. He does not have to change what he is doing because he hears another scholar issuing a fatwa that differs from the one he is following, unless he realizes that what he learned later on is the correct view, on the basis of his confidence in the religious commitment and knowledge of the second Shaykh. (IslamQA)
Well, there are many technicalities involved. I will try to give you a little bird's eye view.
Islam is a beautiful religion and one of its great things is how it has preserved the literature(should i say), and all related details. We all know about quran, that there is no change in it since it was revealed to today, and muslims do believe this book will remain unchanged until the last day.
Difference of opinion comes from the interpretation of Quran, and that interpretation is(mostly) in the light of hadees. Now hadees is also recorded very well, and there is whole hadees science for the understanding of how the authenticity of ahadees is checked and how do we resolve any(apparent) conflicts in ahadees. SOmetimes, if a ruling is based on a certain hadees, and that hadees turns out to be zaeef or weak(technically), than that ruling becomes weak too.
This(hadees sciences) being a very complex and huge field, we see that imams from salaf have different opinions in same matter, while they all provide ahadees and verses for support of that ruling of theirs. Those rulings were given in different areas in different times, and its most probable that one imam might have missed or misunderstood something and gave ruling on that behalf. Now, when a scholar gathers all the rulings about a matter and the basis of that ruling. A scholar can understand and differentiate which opinion is strong, and which opinion is based on misunderstanding(or misinformation maybe).
One thing we need to keep in mind is, that by misunderstanding we don't mean to disrespect any imam from our salaf, in-fact we look up to those imamas in our matters because they have done so much research for us and gathered everything related to that topic. But, one of those opinions by different imams have to be stronger. And we will follow the one which seems stronger. An imam can have more strong opinion for one thing and a weak opinion on other, it really does not matter, we will just follow the seemingly strongest opinion on base of the supporting references from quran/hadees/aasaar etc.
As an example, in the world of science, there are many theories at a time about same thing/phenomena. And it's highly likely one theory makes more sense and for one person and other makes sense for someone else. This 'MAKING SENSE' is of-course based on some rules of science and logical reasoning that fi this is so and so than this should be so. Similarly when we have different opinions on same matter from different sources(imamas), a scholar will evaluate them all combining all the related references(verses/ahadees/aasar) from all sources and then decide what makes more sense.
Also one major consideration when evaluating all the opinions on something is to see to which conclusion most of the scholars tend to be on. And that conclusion has more chances to be right, because most of the scholars ended up with that same conclusion. But, still, that opinion might not be the strongest(most of the scholars might have missed something important, it's possible).
Last thing i should mention, is asool(اصول/rules), there are different asool followed by different sects. Some asools very minorly, and some vary to a great extent. One sect might consider accepting 'hardly weak ahadees' for specific kind of matters, while other sect might not accept any weak hadees at all in any kind of matter. So, when a scholar of one school of thought compares all rulings on a matter, he will surely use the usool he follows and believe are more preferable. And preferable ruling based on those asools might be different for different sects.
All in all, all muslims are trying to follow the true message of islam, mostly they are following a sect or set of asools because they think this is more logical and seems right to them. As far as a muslim is following something not because his father/friend or someone is following that, but because that seems logically right to him, and seems more in accordance to quran and hadees. There is absolutely no issue, and inshallah, everyone will get reward of his efforts.
PS: i intended to write a short note, but ended up writing a blog post instead :D I reviewed the answer and it seems to me, this all information is necessary to understand the topic. Hope it answers your question.