The easiest way to explain it is just to say that in English the word "morning" has two different meanings.
A dictionary gives these as the first definitions of "morning." Notice that definition #1 actually includes two different meanings!
1. The first or early part of the day, lasting from midnight to noon or from sunrise to noon.
2. The dawn.
("Dawn" is the beginning of daylight, a little before sunrise).
By convention in the US, and I think everywhere, each new calendar day begins at midnight. As I write this it is 8:23 p.m, August 11th. A few hours from now it will be 11:59 pm, August 11th. At midnight the date will change from August 11th to August 12th.
As noted, "morning" can refer to the first part of the calendar day, the part that begins at midnight.
"The wee hours of the morning" means the hours of the morning that are represented by small or "wee" numbers like 1, 2, and 3. Thus 2 am is one of the "wee hours of the morning."
"Three O'Clock in the Morning" was the title of a popular song of the 1920s. The lyrics explain the situation:
"It's three o'clock in the morning
We've danced the whole night through
And daylight soon will be dawning
Just one more waltz with you."