Let me put it this way: If Joe says: "there is a deity in the universe", and Jack says: "there is no deity in the universe", let me ask you, who has the heaviest burden of proof? Because in order to conclude beyond question that there is no deity in the universe, Jack must scan the entire universe to make good on his claim. As for Joe, if he finds a deity lurking on the nearest planet, his search for evidence is over. Therefore, the heaviest burden of proof falls on atheists, not on theists.
I think there is a difficulty here. Let first focus on the first part of the above paragraph and suppose it reads as:
Joe: There is a X
Jack: There is no X
Obviously how likely we evaluate those statements to be depend on what X is. If for instance X is "A small smooth rock" Joe is very likely correct, if on the other hand "X" is "Santas magical workshop" the burden of proof clearly rests on Joe.
Normally, in science, if X is something belonging to some new type of object not previously seen the convention is to doubt the existence of X until new evidence comes in. Notice the christian god is about as different from any other thing we know of as can be.
This brings me to the second part of the paragraph. The problem is that how easy it is to confirm (or rule out) the existence of X should not affect our belief if X exists. Take this example
X1 : There is a small magical dwarf in my attick
X2: There is a small magical dwarf on Mars
It is very easy to confirm or rule out X1 and very hard to confirm or rule out X2, but we would normally say both claims were equally unlikely and any person who wished to believe them should have good evidence.