How it is determined that a particular gene or set of genes is responsible for a trait? Is it just a statistical analysis of people showing and not showing the trait and the genes they have (or don't have) and their variants (alleles)? Or, there is something more.
In the below list of technics, I am assuming we are talking about haploid individuals (to avoid discussion about dominance relationship) and assuming no epistasis but the technics can 'easily' be extended to include these cases. I also assume we are talking about coding regions.
Note that I am more of a biostatistician / population geneticist than a molecular geneticist, so I am pretty sure I will unwillingly bias my list toward what I know most. There is a lot to say about every single technic but of course, this cannot be covered in a single StackExchange post.
Statistical association (observational study) with existing variance in natural population. Do carrier of a given allele have another trait value than the carrier of the other allele. Possible confounding variables include population structure and linkage disequilibrium.
- Population structure: To deal with this confounding variable, we can make a genetic PCA and include the first two axes as covariate in a type I sum of squares regression. We can also use tools such as STRUCTURE to infer the population structure and then use the inferred groups as a nominal covariate.
- Linkage disequilibrium: We typically consider a number of marker that we try to associate to a specific phenotypic trait. It is impossible to tell whether the observed association is directly caused by the marker we observe or by a tightly linked marker. For further investigations below technics must be considered.
Statistical association (experimental study). It is possible to create the treatment groups by knocking out (or other technics such as the now popular CRISPR technic) specific gene in out in some individuals and look for a statistical association. The advantage of such manipulation is that we can create the optimal level of genetic variance we need to increase our statistical power.
Transcriptome analysis. Making sure the coding region is being expressed in the tissue of interest. E.g. if you expect a given gene to affect myosin tertiary structure, you might want to make sure the coding region is indeed expressed in the muscles and not only in the brain!
Protein action. It is also possible to study the specific action of the protein coded by the coding region of interest. Such studies typically require the study of an entire biological pathway and is often demanding and quite expensive. Here the methodology are diverse but they're discussion is both outside the scope of the question and outside the scope of my knowledge.
Example - Genetic Basis of Color Adaptation in mice
In a comment, you asked
Is there any research paper, review paper or blog which covers entire details how the association and mechanism was established for any of those discoveries?
In most study cases, not all of the above steps are being considered. Also, even if it is the case, such study will be covered by several peer-reviewed papers and not a single one. I don't know of any peer-review paper that explains the whole process for a particular case but those probably exist.
There are documents that explain such process for layman. For example From Darwin to DNA: The Genetic Basis of Color Adaptations by Hoekstra is a standard reading for undergraduate that will interest you. It is easy and pleasant to read. Quoting from this article
In this essay, we will take a journey — into the laboratory and the field — to understand the genetic basis of adaptation in natural populations of mice, and the phenotype we will focus on is the color of their fur.
The gene of interest in this paper is melanocortin-1 receptor (Mc1r). Depending on what level of understanding you want to reach, the article might be a bit too introductory but it refers to some good peer review papers that will allow you to further your knowledge.