Reference

"À la Sainte Luce" is refering to Lucie de Syracuse who is celebrated the 13 of December.

Meaning

The general meaning is about the evenings which are becoming longer (but not the whole day). It is explained in french here and I will translate this sentence :

En effet, si le soleil se lève de plus en plus tard, il se couche également de plus en plus tard. Les journées semblent donc grandir car les soirées s’allongent.

→ Even if the sun is rising later, it is falling later either. Days look like they are longer because the evenings are getting longer.

Date

In the first link I gave, even the english version states that

Her feast day, known as Saint Lucy’s Day, is celebrated in the West on 13 December.

Flea part

A "puce" is here a flea, a very small insect which is not leaping very high in our point of view (without proportions). It indicates a short distance and by extension a short distance of time.

1/ To grow (croître), it means: how little the length of the day varies.

2/ It’s at St Luce’s day, from St Luce’s day.

3/ Luce or Lucie, is Lucie de Syracuse:
https://fr.wikipedia.org/wiki/Lucie_de_Syracuse

About the proverb: https://fr.wiktionary.org/w/index.php?title=%C3%A0_la_Sainte-Luce,_les_jours_croissent_du_saut_d%E2%80%99une_puce

http://www.linternaute.com/proverbe/1720/a-la-sainte-luce-les-jours-croissent-du-saut-d-une-puce/

English page (She is called Lucia of Syracuse)
https://en.wikipedia.org/wiki/Saint_Lucy

Feast: december 13th.
But:
“A Swedish source[5] states that the date of (Winter Solstice, St. Lucia, Lucinatta, Lucia-day, Lussi-mass …) i.e. December 13, predates the Gregorian which implies that “Lucia’s Day” was Dec 13 in the Julian Calendar, which is is equal to December 21 in the Gregorian, i.e. now. Same source states use of the name “Little Yule” for the day, that it was among the most important days of the year, that it marked the start of Christmas month, and that with the move to the Gregorian calendar (in Sweden 1753) the date (not the celebration) “completely lost its appropriateness/significance”.”

From:
https://en.wikipedia.org/wiki/St_Lucy%27s_Day

4/ Yes, “saut de puce” is an expression. It’s not very common, a bit “littéraire”, but if you use it, everyone will understand what you mean.

Example:

“Les avancées dans ce domaine ont fait des sauts de puce.”

https://fr.wiktionary.org/wiki/saut_de_puce

It means a short amount. A short progress.

Besides all the good answers given, let’s also mention the fact that Luce comes from Latin lux and as such is linked to light (and day).