The initial part of your question appears, as of you want to know why "renunciation of fruits of actions" is better than the "meditation". From the comments it appears, that your question is something else.
Nevertheless, "why is it better?" -- there is no absolute answer in Gita. The 4 types of Yoga-s, in preferential order are listed in this post: What is Yoga according to Bhagavad Gita?
"... what exactly is the meaning of renunciation of fruits of actions and how can I perform it?"
- The "renunciation of results" is called "Karma yoga" (the best among all Yoga-s)
- Karma Yoga in turn is the SAtvika way of life
- The people with SAtvika minds know to uphold the Dharma. All these is explained in detail in this answer, in the last section Karma Yoga = SAtvika way = Following Dharma
- Hence, [SAtvika] renunciation is equivalent to following Dharma
- Dharma is defined as per scriptures (Dharma ShAstra-s) for any given culture, region & era
- "Scripture" here is basically a common "contract" among the people on "do-ables" (viz. Dharma); It gives a good approximation on how to run the society in a self sufficient way; It doesn't count on exceptions
All the above bullets are interlinked one by one. So basically "renunciation of results" boils down to follow the rules.
For example (based on your comment), any society's rule-book will say:
- A father's property belongs to his children by default
- Among the multiple children, the property is divided equally if no will in place
- A boy and a girl may get equal/unequal share based on conventions
- If property is disputed, then one should go to court for resolution
- Court verdict is final; If unsatisfied, then appeal in higher courts
If a person follows above, then he is on Karma Yoga.
How to deviate from Karma Yoga:
- I liked his property, so it's mine; I don't care of his children
- I am the youngest, so I should have bigger property share
- I am a man, so give me the bigger share
- If any dispute, then I will hurt my opponents
- If court is not in my favor, then I will adversely possess the property
Above is just 1 set of deviation, but there can be several variations. One can deviate from rule in many ways.
However, the right set of steps is only 1! Hence in Karma Yoga, one acts as of there is only 1 choice. No brain twisting!
"... stories from Puranas that helps understanding this principle ..."
The ancient examples are lord RAma, great Bhishma, guru Drona, Maharshi VyAsa and few more. All these people, always followed the rule book of their society and never took the law in their hands to twist the result. They could have prevented many problems by denial, but they did it because it was ought to be done like a responsibility.
By the way, to follow the SAtvika way, viz. Karma Yoga, viz. renunciation of results, then don't expect to have necessarily a 'happy life', because:
- 'Expecting' good results from 'not expecting' any results -- itself is self contradictory. In other words, if one starts following "renunciation of results", --because-- it gives good results --- would fail the purpose in the beginning itself
- All the Yoga-s finally lead to liberation. So basically by following Karma Yoga, one wants to "quit" the game of life viz. rebirths. It's not about "winning" it. "Win" is a result, and "quit" is the renunciation.
The verse in the question, also has the answer at the last:
BG 12.12 - ... From renunciation, Peace follows immediately.