The relevant section of the Act is Section 1010(e), which comes at the very end of the act in a section about "technical and conforming amendments." This section doesn't do much: it simply moves a paragraph of statutory text from one place in the U.S. Code to another. Specifically, the section takes a paragraph from 39 U.S.C. 3623(d) — a paragraph that has been there since the 1970s — and moves it to 39 U.S.C. 404(c).

So what does this signing statement mean? First, it pretty clearly says that the Administration reads the moved paragraph as having implicit exceptions that track the Fourth Amendment's exceptions to the warrant requirement.

It may be that this signing statement is nothing... On the other hand, it may be that it hints at a program allowing the government to open postal mail under the claimed authority of the AUMF (Authorization for the Use of Military Force (in Afghanistan, et al)).

Orin Kerr provides some details on the Bush signing statement, and many Volokh readers comment.

I have to wonder if this isn't just a product of beaurocracy. Beaurocrat A is cleaning up beaurocratic regulation B and decides that paragraph C would be more logical if it was organized under section D. Congress approves, and the bill goes before Whitehouse beaurocrat E, who has never seen paragraph C before, and adds signing statement F, designed to protect his client, President G, by providing a mundane explanation of how paragraph C doesn't override an existing understanding of Amendment 4. Journalist H reads this signing statement, jumps to conclusion I, and writes sensationalistic story J, which leads bloggers K through V to freak out, and gives Special Agent X an idea.... Eventually the entire country is beset with an Orwellian Nightmare, and its all due, ultimately, to the anal retentive regulatory care of beaurocrat A.

The Volokh Conspiracy - Has the Bush Administration Claimed New Power to Open Mail?