They didn't. The changes of name (there were actually three of them) took place because of changes in their patron. The first change was because the patron died and they got a new patron, the second because the new patron got a new title, and the third because the king took over as the patron. Neither Shakespeare nor any other member of their company had any say in the company's name, which was simply the name (or title) of the patron with "Men" or "servants" behind it.
One thing's for sure: they were never called "Shakespeare's Men." Not only was Shakespeare not in a position socially to be a patron, but he was not even the leader of the company.
It indicates (as the names of all Elizabethan companies did) the name of the patron of the company.
In 1603, The Lord Chamberlain's Men became The King's Men.
The theatre company to which Shakespeare belonged for most of his career started off as The Lord Chamberlain's Men, but when their patron changed from the Lord Chamberlain to the King their name changed to the King's Men.
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Shakespeare's acting company was first known as Lord Chamberlain's Men. The name was changed to The Kings Men in 1603 when King James I ascended the throne and became the company's patron.
The Lord Chamberlain's Men became The King's Men.
They became known as the King's Men when King James took up patronage of the company.
Shakespeare was a shareholder in Lord Hunsdon's Men who changed its name to the Lord Chamberlain's Men, who later became the King's Men.
The Lord Chamberlain's Men became The King's Men in 1603.
They are more commonly remembered as the King's Men.
His company became the King's Men, after James I ascended to the throne.
Lord Hunsdon's Men became the Lord Chamberlain's Men which became the King's Men.