How is the federal government able to be involved in education when the Constitution reserves that right to the states?

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,  href="http://www.law.cornell.edu/supct-cgi/get-us-cite?312+100">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.