Logan Act:
"953. Private correspondence with foreign governments.
Any citizen of the United States, wherever he may be, who, without authority of the United States, directly or indirectly commences or carries on any correspondence or intercourse with any foreign government or any officer or agent thereof, with intent to influence the measures or conduct of any foreign government or of any officer or agent thereof, in relation to any disputes or controversies with the United States, or to defeat the measures of the United States, shall be fined under this title or imprisoned not more than three years, or both.
This section shall not abridge the right of a citizen to apply himself, or his agent, to any foreign government, or the agents thereof, for redress of any injury which he may have sustained from such government or any of its agents or subjects."
In 1975, Senators John Sparkman and George McGovern were accused of violating the Logan Act when they traveled to Cuba and met with officials there. In considering that case, the U.S. Department of State concluded:
The clear intent of this provision [Logan Act] is to prohibit unauthorized persons from intervening in disputes between the United States and foreign governments. Nothing in section 953 [Logan Act], however, would appear to restrict members of the Congress from engaging in discussions with foreign officials in pursuance of their legislative duties under the Constitution. In the case of Senators McGovern and Sparkman the executive branch, although it did not in any way encourage the Senators to go to Cuba, was fully informed of the nature and purpose of their visit, and had validated their passports for travel to that country.
source:
https://en.wikipedia.org/wiki/Logan_Act
http://www.cotton.senate.gov/sites/default/files/150309%20Cotton%20Open%20Letter%20to%20Iranian%20Leaders.pdf
In other words, Congress are public officials, who are not engaging in "private correspondence. In this case, the letter, which is public record, is educating Iran or reminding their officials that any agreement made at the executive level is subject to change. That being an informative statement and offering no bribes, incentives or threats, but merely truthful information about how our country's government operates, should not be cause for alarm concerning a breach of the Logan Act.
Except that the senators who signed are all republican, so we will be hearing about it in the news ad nauseam and from angry leftists who posted here in the original (OP).
Tom Cotton's letter to Iran:
It has come to our attention while observing your nuclear negotiations with our government that you may not fully understand our constitutional system. Thus, we are writing to bring to your attention two features of our Constitution—the power to make binding international agreements and the different character of federal offices—which you should seriously consider as negotiations progress.
First, under our Constitution, while the president negotiates international agreements, Congress plays the significant role of ratifying them. In the case of a treaty, the Senate must ratify it by a two-thirds vote. A so-called congressional-executive agreement requires a majority vote in both the House and the Senate (which, because of procedural rules, effectively means a three-fifths vote in the Senate). Anything not approved by Congress is a mere executive agreement.
Second, the offices of our Constitution have different characteristics. For example, the president may serve only two 4-year terms, whereas senators may serve an unlimited number of 6-year terms. As applied today, for instance, President Obama will leave office in January 2017, while most of us will remain in office well beyond then—perhaps decades.
What these two constitutional provisions mean is that we will consider any agreement regarding your nuclear-weapons program that is not approved by the Congress as nothing more than an executive agreement between President Obama and Ayatollah Khamenei. The next president could revoke such an executive agreement with the stroke of a pen and future Congresses could modify the terms of the agreement at any time.
We hope this letter enriches your knowledge of our constitutional system and promotes mutual understanding and clarity as nuclear negotiations progress.
Sincerely,