What appeal to emotion does Jefferson use in the first paragraph of the Declaration of Independence?

Limiting the answer to the first paragraph alone, Jefferson's use of the phrase "a decent respect to the opinions of mankind" can be construed as an appeal to emotion.

Strictly speaking, the Declaration of Independence could have been just that: a plain legal assertion that the American colonies represented by the Declaration's signatories were no longer under English rule.

Instead, by asserting that "a decent respect to the opinions of mankind" requires that the text...

Limiting the answer to the first paragraph alone, Jefferson's use of the phrase "a decent respect to the opinions of mankind" can be construed as an appeal to emotion.

Strictly speaking, the Declaration of Independence could have been just that: a plain legal assertion that the American colonies represented by the Declaration's signatories were no longer under English rule.

Instead, by asserting that "a decent respect to the opinions of mankind" requires that the text enumerate the complaints of the colonists, Jefferson employs an "appeal to consequences," a subset of emotional appeal. In effect, he's saying that unless the Declaration details exactly why the Colonies are declaring independence from England, they risk being deemed unjust or unreasonable by the rest of the world. 

That appeal sets up the rest of the Declaration. The second paragraph of the Declaration tends to be the part Americans remember. It's the bit with the beautiful assertions of "all men are created equal" and "life, liberty and the pursuit of happiness."

The rest of the Declaration is rather different. It consists in large part, to use its own language, of "a long train of abuses and usurpations" that the authors attributed to the English crown. By appealing to consequences in the first paragraph, Jefferson and the Declaration's other contributors give themselves license to list various terrible things they felt were inflicted on them by English authority. In doing so, they hoped to cast themselves as the injured party in the conflict and justify their actions on the global stage.

To state the obvious, it worked. America won the War of Independence, in large part due to foreign aid, particularly from France. By all appearances, Jefferson's appeal to emotion did its job admirably.