How we cite our quotes: (Article.Sentence)
Quote #1
The aim of all political association is the preservation of the natural and imprescriptible rights of man. These rights are liberty, property, security, and resistance to oppression. (2.1-2)
They're explaining what the role of the government should be, and at the same time setting up the aim of the government they wished they had. They're also listing the basic rights that all the others stem from.
Quote #2
Law can only prohibit such actions as are hurtful to society. Nothing may be prevented which is not forbidden by law, and no one may be forced to do anything not provided for by law. (5.1-2)
This entire Declaration is about setting boundaries and trying to guess how the monarchy and nobility might try to cross them. Here they're explaining what the law can and cannot do and daring anybody to challenge it.
Quote #3
Law is the expression of the general will. Every citizen has a right to participate personally, or through his representative, in its foundation. (6.1-2)
When this was written France didn't have a true representative government, but they're suggesting that creating one is essential. This Declaration is super-hopeful about the future of France; these guys were optimists.
Quote #4
All the citizens have a right to decide, either personally or by their representatives, as to the necessity of the public contribution; to grant this freely; to know to what uses it is put; and to fix the proportion, the mode of assessment and of collection and the duration of the taxes. (14.1)
Again, it's all about citizens participating: a totally new concept to France where the king and his advisors had been running the show for…well, for basically ever. The representatives of the Assembly wanted to change that; they wanted to try running things for a while.
Quote #5
Society has the right to require of every public agent an account of his administration. (15.1)
This is an example of them anticipating what might go wrong and planning ahead. They know that the tax-free nobles are crafty and they want to outfox them by demanding upfront that they account for themselves.