Back in the olden days, back where there were houses made of sod on the American prairie that people actually lived in, about the time that Laura Ingalls Wilder was still Laura Ingalls, ladies used to wear dresses, men wore trousers (or pants). Also, ladies were treated as second class people. Ladies could not own property of their own. Ladies could not vote. Ladies could not enter into contracts. If you wanted to get someone in a household that had the ability to enter into a binding agreement, you needed a man. The only way you could tell a man from a woman in those days was to check to see what they were wearing. If the person was wearing pants, it was a man, and you could do serious business with him. If the person was wearing a dress, you did not bother with them.
Thus, the person in a household who makes decisions and is otherwise in charge "wears the pants".
According to phrases.org, this idiom was well known in the late 19th century. The first known appearance in print was in an November 1880 article in The Manitoba Daily Free Press where the phrase "wear the trousers" was used.
An answer at answers.com quotes American Heritage Dictionary of Idioms to the effect that to not give someone the time of day means
Ignore someone, refuse to pay the slightest attention to someone, as in He's tried to be friendly but she won't give him the time of day. This expression, first recorded in 1864, alludes to refusing even to answer the question, "What time is it?"
By contrast, an answer at tywkiwdbi.blogspot.com says:
the expression goes far back beyond the time when people wore watches... In Shakespeare's day, the meaning was quite clear. "Good time of day" or "fair time of day" was a salutation just like "good morning" or "good evening"...We no longer greet people by saying "good time of day," but we still use the idea of giving such a greeting as a sign of favorable attention. In other words, refusing to give someone the time of day is thinking so little of him that you would not say hello to him on the street." (credit to Scribal Terror)
If tywkiwdbi and Scribal Terror are right, then the expression is rather older than the AHD Dictionary of Idioms said.
Edit: The BBC Learning English presentation “Not give someone the time of day” says:
Long ago, in Shakespeare's time, the phrase 'good time of day' was a greeting often used. These days we say 'good morning' [or 'good afternoon'] … So to say that you wouldn't give someone the time of day means you wouldn't want to greet them or say hello. So the saying means you refuse to give someone your attention.
As an example of the phrase used in Shakespeare's time, consider King Henry VI, part II, Act III, scene I, as Queen Margaret speaks:
Can you not see? or will ye not observe
The strangeness of his alter'd countenance?
...
We know the time since he was mild and affable,
...
But meet him now, and, be it in the morn,
When every one will give the time of day,
He knits his brow and shows an angry eye,
And passeth by with stiff unbowed knee,
Edit 2: Another example from Shakespeare (as pointed out by ΜετάEd's reference) is from King Richard III, Act I, scene III, when Buckingham says “Good time of day unto your royal grace!”. Note, Shakespeare is believed to have written both plays about 1591; he might or might not have put 16th-century speech into the mouths of 15th-century people. Henry VI was King of England and/or France at various times between 1422 and 1471, and Richard III from 1483 until his demise in 1485.
Best Answer
OP is mistaken about the exact meaning of the expression. It's not about acquiring "new" skills, but about how you got started on the skills you already have. Here's a definition from dictionary.com
It's a metaphoric reference to when a baby's teeth first appear. They grow (cut) through the gums - often painfully, which also gives us the figurative usage teething troubles.
Once the baby has "cut its teeth", it's properly equipped for the all-important "real-world" task of chewing solid food (metaphorically, for tackling more complex problems in professional life, etc.).
As that dictionary example suggests, the expression is often used boastfully/facetiously (in reality the hunter probably started with much less challenging prey, such as grubs, mice, and rabbits).