The second sentence is correct and can be shorten to:
What do you understand by 'better life'.
Another example:
I understand by God's being eternal that he always has existed and always will exist.
Some words needs a preposition (by, to, of, from ...) to take an argument and if you omit that preposition the sentence become ungrammatical. Specially, if they take more than one argument:
You say:
The Police requested a grant from the government.
You can't say:
*The Police requested a grant the government.
Understand takes a direct object without any preposition like:
I can understand the lesson.
I understand that the God always has existed and always will exist.
In your sentence "what" plays this role and "What do you understand" is a complete sentence, but for its other argument it needs "by" preposition
What do you understand by 'democracy'.
I understand by 'democracy' that everyone has the right to to take part in the government of his country,
I understand that everyone has the right to to take part in the government of his country.