It is by no means a fact that the executive branch is the strongest branch of the American government. You could argue that it is, but you could also argue that it is not. After all, the president has fairly limited powers. The most important example of this is the fact that he (or someday she) cannot make laws. However, we can argue that the executive branch is the strongest branch.
One reason for this is that the executive branch carries out all the laws. If the executive branch decides that it does not want to enforce a law very strictly, it can probably get away with that unless Congress really decides to work hard to force it to do so. This is how, for example, President Obama can say that the US government will not prosecute marijuana growers in states where marijuana is legal.
Another reason for this is that the executive branch can veto bills. Today, we know that the Republicans in Congress would probably be able to get enough votes to repeal Obamacare. However, with President Obama in the White House and able to veto laws, there is no way such a law could ever be passed.
A third fact is that the executive has almost complete power over foreign policy. The president can negotiate treaties, though the Senate does have to ratify them. The president can command the military without Congressional approval. The president has almost unlimited ability to do what he or she wants in this area.
The most important reason why the executive branch is strong is because the president is the most visible leader for the nation. Only the president is voted for by people from every state. Most people know who the president is, but only relatively few people can name many members of Congress. When the president gives a speech, it is big news but when a member of Congress gives a speech it is not. Therefore, the president has much more ability to lead the nation and to try to persuade people to support certain laws.
In these ways, it is at least possible to argue that the executive is the strongest branch of government.