Why should we stop the Samsara? Throughout the holy texts it is implied that the reader is looking for a way to stop the cycle.

Why should the reader want to stop the cycle?
I think that dukkha is the reason why the reader is looking for a way to stop the cycle.

I think that, because the "four noble truths" were the first thing that the Buddha explained, during his first sermon after enlightenment (with dukkha being the subject of the first of these four noble truths).

The fact that dukkha is the cause of the search (i.e. that "the search is the result of stress") is also stated explicitly, in the Nibbedhika Sutta:


  "'Stress should be known. The cause by which stress comes into play should
  be known. The diversity in stress should be known. The result of
  stress should be known. The cessation of stress should be known. The
  path of practice for the cessation of stress should be known.' Thus it
  has been said. In reference to what was it said?
  
  Birth is stress, aging is stress, death is stress; sorrow,
  lamentation, pain, distress, &amp; despair are stress; association with
  the unbeloved is stress; separation from the loved is stress; not
  getting what is wanted is stress. In short, the five
  clinging-aggregates are stress.
  
  "And what is the cause by which stress comes into play? Craving is the
  cause by which stress comes into play.
  
  "And what is the diversity in stress? There is major stress &amp; minor,
  slowly fading &amp; quickly fading. This is called the diversity in
  stress.
  
  "And what is the result of stress? There are some cases in which a
  person overcome with pain, his mind exhausted, grieves, mourns,
  laments, beats his breast, &amp; becomes bewildered. Or one overcome with
  pain, his mind exhausted, comes to search outside, 'Who knows a way or
  two to stop this pain?' I tell you, monks, that stress results either
  in bewilderment or in search. This is called the result of stress.




Saying that dukkha is the reason for the search might seem like a one-size-fits-all answer: if several/many people want to stop the cycle, do they all have the same reason for wanting to?

For that reason, dukkha is a broad term with apparently several meanings or usages; the WIkipedia article starts with ...


  Dukkha (Pāli; Sanskrit: duḥkha; Tibetan: སྡུག་བསྔལ་ sdug bsngal, pr. "duk-ngel") is a Buddhist term commonly translated as "suffering", "anxiety", "stress", or "unsatisfactoriness".[a] The principle of dukkha is one of the most important concepts in the Buddhist tradition. The Buddha is reputed to have said: "I have taught one thing and one thing only, dukkha and the cessation of dukkha."


... and then continues, with many various categories (and translations) of dukkha.