Spokesman Josh Earnest made the announcement at yesterday's press briefing, just as the Senate Foreign Relations Committee was convening for what would be a 19-0 vote in favor of the bill. The declaration reversed months of explicit administration threats to veto legislation that would give Congress an expanded voice in Iran negotiations.
The White House spin was that Corker-Menendez had been substantively hollowed out by a morning compromise between Sens. Corker and Cardin, so that had become acceptable. The bills' supporters countered that the core oversight requirements remained intact, and that the administration caved because it knew a veto would be overridden. Corker on Twitter: "The simple fact is that the White House dropped its veto threat because they weren’t going to have the votes to sustain a veto" .
The subsequent news cycle did not reflect the White House's messaging:
Even the New York Times editorial board, which was very unkind to the Senators, read the politics in a way that cut against the White House's messaging: The committee’s action gives momentum to those who have bitterly criticized Mr. Obama for negotiating with Iran... Mr. Obama initially threatened to veto the legislation, but he backed off rather than face a bipartisan override of his veto.  The next step on the Senate side is for the bill to be brought to the floor, where there will be an amendment process and then a vote. On the House side, lawmakers will have to craft parallel legislation, but The Hill reported last nigtht that Democrats are already jumping on board so the process is likely to be quick .
- Reuters: In setback, Obama concedes Congress role on Iran deal: "U.S. President Barack Obama conceded on Tuesday that Congress will have the power to review a nuclear deal with Iran, reluctantly giving in to pressure from Republicans and some in his own party after they crafted a rare compromise demanding a say." 
- Daily Beast: Obama Blinks on Iran Nuke Vote: The White House just did a 180 on a controversial bill to let Congress vote on the Iran nuclear deal. It’s a big win for the deal’s critics: "In the standoff with Congress over the Iranian nuclear deal, President Obama just blinked. Faced with the prospect of a backlash from members of Obama’s own party on his signature foreign policy initiative, the White House on Monday said it’d be willing to sign a bill that will prevent the administration from lifting sanctions on Iran while Congress reviews whatever final deal is reached with Tehran over its nuclear program." 
- New York Times: Obama Yields, Allowing Congress Say on Iran Nuclear Deal: "The White House relented on Tuesday and said President Obama would sign a compromise bill giving Congress a voice on the proposed nuclear accord... White House officials insisted they extracted crucial last-minute concessions. Republicans - and many Democrats - said the president simply got overrun." 
- Times of Israel: Veto override fear spurred Obama to back oversight bill, groups claim: From Christian Zionists to American Iranians, nobody’s buying that compromise made Iran legislation palatable to White House: "The White House said Tuesday its decision to back a bill increasing Congressional oversight on any nuclear deal with Iran was due to concessions, but many in Washington think the about face is more about adding up veto-busting votes than the subtraction of controversial measures." 
At issue is how easily the will of Congress can be blocked, based on how the Menendez-Corker bill has been formulated:
As the editors of The Wall Street Journal analyzed it, “The majority could offer a resolution of disapproval, but that could be filibustered by Democrats and vetoed by the President. As few as 41 Senate Democrats could thus vote to prevent it from ever getting to President Obama’s desk—and 34 could sustain a veto. Mr. Obama could then declare that Congress had its say and ‘approved’ the Iran deal even if a majority in the House and Senate voted to oppose it.”If so, Obama lost nothing by his "concession" to Congress. And he knows it.