Quite simply, Shakespeare's acting troupe was called "The Lord Chamberlain's Men" and was later called "The Kings Men." The reason for the change is what calls for more detail.
Originally named "The Lord Chamberlain's Men," the group was formed under the patronage of Lord Carey (which means that it was Lord Carey who provided them funding to produce plays). Lord Carey's title at the time was Lord Chamberlain, hence the name of the troupe after their patron: "The Lord Chamberlain's Men."
It was founded during the reign of Elizabeth I of England in 1594, under the patronage of Henry Carey, 1st Baron Hunsdon, then the Lord Chamberlain, who was in charge of court entertainments.
At this time, Shakespeare was considered one of the founding members and paid quite handsomely. In "The Lord Chamberlain's Men," Shakespeare was among his contemporaries also known for their acting abilities such as Burbage and Kemp. They were all shareholders. This was at the very end of the sixteenth century.
However, very early in the seventeenth century, namely 1603, England had a new king who was a very big fan of Shakespeare's acting troupe; therefore, the king himself (James I) became their one an only patron. For this reason, they changed the name of the acting troupe to "The King's Men." During this time period, Shakespeare wrote ferociously and, in fact, almost a dozen plays were put on in court within one year!