Who first figured out that Earth's seasons are caused by the tilt of
the Earth, and in or around what year did they do it?

Although used in earlier time, such as for sun dials...the
quantification of the tilt of the earth seems to be answered here
(astronmer at Cornell):
http://curious.astro.cornell.edu/question.php?number=245

Who (and when) discovered that the earth's axis is on a 23 degree tilt?

Astronomers define a coordinate system on the sky that lets them
identify the relative positions of stars, planets, the Sun and other
celestial phenomena (the coordinates are called "right ascension" and
"declination", but we don't need to worry about that here). Because of
their great distance from the Earth, the positions of most stars are
fixed on the grid. The positions of the planets and the Sun, however,
are seen to "travel" through the coordinate grid in systematic (though
sometimes complicated) patterns. The paths of the planets was
carefully observed by Tycho Brahe in the late 1500s. Following Brahe's
death in 1601, Johannes Kepler used the data to develop a set of laws
that govern the way the planets move around the Sun (he proposed them
in 1609). These laws, (creatively) called Kepler's Laws, still hold
today.

With Kepler's laws in hand, one can easily interpret the motion of the
Sun through the coordinate grid described above as the result of the
tilt of the Earth by 23.5 degrees with respect to the plane in which
the Earth orbits the Sun. Since Kepler was the one to correctly
interpret the motion of the planets and the Sun through the sky, he is
the one that I would attribute the title "Discoverer of the 23.5
degree tilt", in the early 1600s.

Here, for what it is worth, is the wiki link:
http://en.wikipedia.org/wiki/Johannes_Kepler

Also books on Google Books:
http://books.google.com/books?as_q=&num=10&ie=UTF-8&btnG=Google+Search&as_epq=&as_oq=&as_eq=&as_brr=0&as_vt=&as_auth=Johannes+Kepler&as_pub=&as_drrb=c&as_miny=&as_maxy=&as_isbn=

With regards to the seasons and tilt specifically...
The procession of the Equinoxes was quantified by :
# Discovered by Hipparchus of Nicaea (c. 150BC), but may have been
known to the Babylonians.

Source: http://www-astronomy.mps.ohio-state.edu/~pogge/Ast161/Unit2/seasons.html

Wiki:
http://en.wikipedia.org/wiki/Hipparchus_of_Nicaea

Thanks for the help, but I had this information already. It's not the
degrees of tilt I'm interested in, but rather the revelation that the
tilt causes the seasons. The Babylonians sounds more reasonable, but
that is what I'd like to narrow down. No doubt that Kepler and
Hipparchus are correct, but not quite what i'm looking for. Thanks for
the input.

http://earthobservatory.nasa.gov/Library/Giants/Milankovitch/milankovitch_3.html
I believe you will be interested in reading about Serbian astronomer
Milutin Milankovitch. He may not have been the first to recognize the
relationship, but he was able to correlate seasonal variations
directly to changes in the tilting of the planet's axis. His data is
still used today to correlate seasonal variations to geological time.

GENESIS 8;22 [ . 22 For all the days the earth continues, seed sowing
and harvest, and cold and heat, and summer and winter, and day and
night, will never cease.?]. From the Bible indicates that it was known
right from Moses' time. Certainlt the early Egyptians knew a whole lot
of astronomy. Their pyramids are based on angles and alignments. So
were the druids of Stonehenge fame.

Thanks again for the help. The excerpt from the Bible, however, shows
that the seasons are recognized, but nothing regarding their
relationship to the tilt of the Earth.

You asked who figured out the seasons are due to the tilt of the
Earth.  I think the answer depends on what you mean by ?tilt?.  The
normal meaning, that the Earth?s spin axis is 23.5 degrees from its
orbital axis, really can?t be given unless one realizes that the Earth
goes around the Sun.  As the Earth orbits, the tilt is preserved by
conservation of angular momentum and that means the days are longer
when the axis points toward the Sun; during the season we call Summer.
 Thus, with this meaning of tilt, the answer would have to be Kepler
in the early 1600?s; unless the Greeks (who early-on theorized the
Earth circles the Sun) wrote on this.  In fact, it may well be those
Greeks since the answer is obvious once you have the Earth orbiting
the Sun.  However, the fact that days in summer are longer because of
the ?tilt? of the Sun?s orbit through the sky was known since
antiquity (before the Babylonians), so if you mean that by tilt then
this knowledge was pre-historical.  (All farmers know how the Sun
moves through the sky during the different seasons; its motion is
?tilted?.)