How did Louis Armstrong get the name Satchmo?

Because he had a large mouth. Short for "satchel mouth".

"King Oliver and other early jazz musicians called Louis "Dipper Mouth" or "Satchel Mouth", presumably because of his large mouth. Early in the thirties, Louis visited England and was given his trademark handle "Satchmo" when British fans heard the "Satchel Mouth" tag incorrectly. Billie Holiday called him "Pops." Gate was also used to designate Armstrong at some point."

http://www.allaboutjazz.com/php/article.php?id=1403

The nicknames Satchmo and Satch are short for Satchelmouth. Like many things in Armstrong's life, which was filled with colorful stories both real and imagined, many of his own telling, the nickname has many possible origins.

The most common tale that biographers tell is the story of Armstrong as a young boy dancing for pennies in the streets of New Orleans, who would scoop up the coins off of the streets and stick them into his mouth to avoid having the bigger children steal them from him. Someone dubbed him "satchel mouth" for his mouth acting as a satchel.

In 1932, Melody Maker magazine editor Percy Brooks greeted Armstrong in London with, "Hello, Satchmo!", and thence the nickname took root.[citation needed]

Early on he was also known as Dipper, short for Dippermouth, a reference to the piece Dippermouth Blues.[36] and something of a riff on his unusual embouchure.

https://en.wikipedia.org/wiki/Louis_Armstrong