The federal government is able to involve itself in education because it is not simply dictating to the states what they must do. Instead, it is telling them what they must do if they want federal aid. This is the important legal distinction.
The federal government gives states large amounts of money to help fund their educational systems. The states have no constitutional right to this money. Instead, the federal government has the right to set whatever conditions it wishes on the money. This is why laws like No Child Left Behind are constitutional. The states have a choice. They can comply with the law and get money from the federal government or they can decline to comply and forego the money. They are not simply being told what to do.
Because the federal government is just putting conditions on money it give the states (rather than simply ordering the states to do certain things), it is allowed to involve itself in education.
New York v. US, 488 U.S. 1041 (1992) basically provides that where the federal government is willing to pay for an inititave it wishes to impliment, it can financially incentivize cooperation of state legislatures.
I assume that you are determining that education is a power reserved to the states because it isn't explicitly mentioned in the constitution and you are literally reading the 10th ammendment. The Tenth ammendment is a bit more nuanced than it appears on its face though:
"It is in this sense that the tenth ammendment "states but a truism that all is retained which has not been surrendered." United States v. Darby, 312 U.S. 100, 124 (1941). As Justice Story put it, "[t]his amendment is a mere affirmation of what, upon any just reasoning, is a necessary rule of interpreting the constitution. Being an instrument of limited and enumerated powers, it follows irresistibly, that what is not conferred, is withheld, and belongs to the state authorities." 3 J. Story, Commentaries on the Constitution of the United States 752 (1833). This has been the Court's consistent understanding: "The States unquestionably do retai[n] a significant measure of sovereign authority . . . to the extent that the Constitution has not divested them of their original powers and transferred those powers to the Federal Government." Garcia v. San Antonio Metropolitan Transit Authority, supra, at 549 (internal quotation marks omitted)." quoting New York v. US 488 U.S. 1041 (1992), at http://www.law.cornell.edu/supct/html/91-543.ZO.html.
In reality, the only power that was explicitly understood by the framers to be reserved to the states is police power (the authoriy to protect the health, safety, and morality of individuals). That isn't an absoloute authority, but generally police power is what the tenth ammendment refers to.
The answer dates back to the Sixteenth Amendment, which removed the constitutional prohibition against the federal government taxing Americans directly. The constitution had provided for a direct tax, but only one that was distributed fairly across the states based on census results. The Amendment allowed the federal government to fund itself without going through the states. Having begun taxing the citizens directly, it was only a matter of time before politicians began returning that money to the citizenry in politically targeted ways: one of these ways was the federal funding of public education.