Explain why triacylglycerols migrate faster than phosphoglycerides in
thin-layer chromatography on silica gel (stationary phase) with a
predominantly hexane solvent (mobile phase)

The suggested url given by jlchem-ga is very helpful and would be a
great place to start. It does give introductory information on what
polarity is and how thin layer chromatography works. It also has a
good history of chromatography and how it relates to thin layer
chromatography (TLC).

However, it doesn't really seem to answer the question you are asking.
After also consulting a book by Morris Kates, Techniques in
Lipidology, Laboratory Techniques in Biochemistry and Molecular
Biology, Volume 2, part2, 1986, Elsevier (ISBN= 0-444-80737-3)I have
an answer for you.

TLC is basically a chromatographic technique that involves the
application of a thin layer of a stationary phase (usually silicic
acid which is usually acid treated Florisil). The TLC plate is placed
in a chamber containing the mobile phase (usually some combination of
hexane, ethyl ether, acetic acid). A sample containing
triacylglycerols and phosphoglycerides can then be applied near the
bottom of the plate, dried, and then placed in the chamber containing
the mobile phase. The lipids in the sample bind to the silicic gel
primarily by ionic interactions. Hydrophobic interactions are also
involved but not to as great an extent. The triacylglycerols being
less polar than the phosphoglycerides are less tightly bound to the
stationary phase. The mobile phase is quite hydrophobic and when it
interacts with the sample the hydrophobic triacylglycerols are more
likely to be freed from the stationary phase and will move more
rapidly up the plate than the more polar phosphoglycerides. The
phosphoglycerides are more polar because the phosphate attracts the
nearby electrons more strongly than the hydrogens and carbon resulting
in a portion of the molecule that is negative (the phosphate) and
another portion that is positive (the rest of the phosphoglyceride
molecule). Due to their more polar nature they are less likely to be
released from the stationary phase than the triacylglycerides.
However, phosphoglycerides are somewhat hydrophobic as well and will
move with the mobile phase but not as rapidly as the more hydrophobic
triacylglycerols will. As a result when the TLC run is finished the
more hydrophobic triacylglycerides will have migrated further than the
more polar phosphoglycerides.

Lipids are carried up the silica gel by the mobile phase, and using a
predominantly hexane solvent system will cause the less polar lipids
to travel further than the more polar lipids.

The phosphate group makes phosphoglycerides more soluble in hexane, so
they aren't "pushed" as strongly as the triacylglycerols.

http://207.198.77.35/pdfs/experiments/113.pdf has a decent tutorial on
why TLC works.

Enjoy