How to use hook_form_search_block_form_alter?
So What is the method to connect this all dots, from Hook name to what can we change with that hook.

There isn't one - each hook is different, has different input, different output, different contents, affects different parts of the system, and so on. You have to look for documentation contextual to what you're working with, each time you have something different.
In this case, you're working with a form, which means you should be referring to the Form API. From there it's an easy search to find information about #attributes:

Description: Additional HTML attributes, such as 'class' can be set using this mechanism.
Values: Any HTML attribute not covered by other properties, e.g. class (for control types), enctype (for forms).

It's basically an arbitrary pass-through of the key/value, so
$form['search_block_form']['#attributes']['foo'] = 'bar';

Will get you (e.g.)
&lt;input type=&quot;text&quot; foo=&quot;bar&quot; /&gt;