Why does "trip the light fantastic" mean "to dance"?

Solution 1:

I lighted upon the following excerpt, and thought it explained the meaning of this charming and idiosyncratic phrase quite well. From Etiquette, politeness, and good breeding: embracing all forms and ceremonies in the etiquette of marriage, etc. Printed in London, 1870.

But one, on whom a brighter light than he possessed dawned in far-off years, enjoined the utmost refinement both in dress and manner to the daughters of his people. Remember this when “tripping on the light fantastic toe,” and preserve the strictest modesty in all your movements. Remember also that it is your safeguard, throwing a halo of light and purity around you; and therefore do not affect to be an accomplished dancer, to display the science and agility of an artiste. It is sufficient that you dance with ease and grace—that you enter into the amusement as becomes a gentlewoman, neither with careless indifference, nor yet with affectation or excessive hilarity. Carefully avoid all such dances as are offensive to refinement and good taste: [...]

In other words, tripping refers to movement; to being nimble-footed. Oxford Dictionaries defines trip as: Walk, run, or dance with quick light steps. The idea of running quickly also appears in Shakespeare's The Tempest (1610–11):

Ariel:
Before you can say ‘come’, and ‘go’,
And breathe twice; and cry, ‘so, so’;
Each one tripping on his toe,
Will be here with mop, and mowe.¹
Do you love me, master, no?

Several sources online suggest that John Milton drew inspiration from this Shakespearian line when he wrote his poem L’Allegro, published in 1645 and included the following verse

Come, and trip it, as you go,
On the light fantastic toe;

The light fantastic modifies toe, which symbolises feet, and means to dance in an agile, effortless and fantastic manner. A similar expression, which should help to understand further, is to be light on one's feet.

Solution 2:

The common reference (as mentioned for instance in wordreference is from a verse in Milton’s L’Allegro (a pastoral poem by John Milton published in 1645).
The Wikipedia article list all the other references/origins.

"Come, and trip it as you go, on the light fantastic toe."

This article illustrates the result when it is presented as modern ballet:

L’Allegro, il Penseroso ed il Moderato is uncategorisable - it’s both dance and oratorio, both narratively coloured and yet at times more formally complex than purest ballet.

Milton's phrase, 'Trip it as you go/ On the light fantastic toe', could have been minted for Mark Morris.

Gramophone and culturevulture concur:

There is a verse in Milton’s L’Allegro that may have inspired Morris to craft this highly innovative dance: "Come, and trip it as you go, on the light fantastic toe."
His dancers can trip the light fantastic with the best of them [...].