Following the Enlightenment tradition of thought, the framers believed that a divided government predicated on the notion of checks and balances would be the best way to ensure that government does not become a tool that moves too far away from the wishes of the people. The creation of branches of government that have specific and different functions with the ability of each branch to limit the other two helps to create a government where little bad can be done because of the presence of other checks or limitations to institutional power. In the end, the power that each branch might not be exactly equal for in different contexts power varies, but the system conceived helps to create the branches as elements that have to work together to complement one another. It can also be a system where nothing, good or bad, becomes readily accomplished because of the different checks that are present.
When the three branch system and checks and balances were designed at the Constitutional Convention in 1787, you have to remember that only four years earlier had we finally won independence from Britain. The Founding Fathers were afraid of a system that would devolve into another monarchy - an American version of King George III, so they wanted a system that would protect against that.
At the same time, they had had the Articles of Confederation for several years and it was clearly an ineffective form of government too, with almost no central authority at all. Shay's Rebellion proved this. So we had to have a government that was able to run a nation, defend it, and get things done if we were to survive.
The Separation of Powers and Checks and Balances compromises then were designed to do these two things for us - make us more functional but still protect against a future King.
Well, the short answer is that the founding fathers designed it that way. Of course you also have numerous other arguments for one being more powerful than the others, but originally it was designed to check each other.
For example, the president theoretically would have to get Congress' approval before sending troops overseas. This has been cricumvented in a variety of ways, both by using the War Powers Act as well as simply not defining a conflict as a war and sending troops to "police actions" such as Korea or Vietnam. This might be one example where you could say that the three branches are not quite equal given current political feelings.
You also have the Supreme Court which is supposed to be able to check both of the other branches by declaring actions of either the Executive or the Judicial branch as unconstitutional. But again, you can argue that this has in some ways been compromised by the ability of either a president or a congress to filter certain candidates for judgeships out so that they determine what the supreme court will or won't do.
I am not sure that you really can say that all of the three branches of government are equally powerful. But if I had to argue that they were, I would say that it is because each can sort of cancel out an action of the other.
For example, the President can veto a law that Congress has passed. In turn, Congress can override the President's veto if it has enough votes.
When it comes to the judicial branch, they can say that a law passed by Congress or an action of the President in unconstitutional.
In that way, each branch can cancel out things the others do. This, you can argue, makes them all equal.