He became their patron.
Lord Chamberlains Men
They were the acting company with which he was associated for nearly twenty years.
The Lord Chamberlain's Men were the acting company which William Shakespeare belonged to. He performed with them, and his plays were the property of the company. The connection of the company with Shakespeare is what makes it significant.
They were a theatre company. They put on plays.
The acting company to which Shakespeare belonged for most of his working life was called The Lord Chamberlain's Men from 1594-1603, and The King's Men after that. Shakespeare was not the troupe's leader, but rather a partner.
They are more commonly remembered as the King's Men.
There is no hard evidence to prove which companies Shakespeare worked with prior to helping to found The Lord Chamberlain's Men. The ususal suspects are Queen Elizabeth's Men, Pembroke's Men, Derby's Men, and Strange's Men, but that is just surmise.
Lord Chamberlains men
The name of every Elizabethan and Jacobean acting company tells you the name of the company's patron or sponsor. It's like having an acting company called the Coca-Cola Players. Lord Strange's Men, The Lord Admiral's Men, The Lord Chamberlain's Men and the King's Men were sponsored by Lord Strange, the Lord Admiral, the Lord Chamberlain, and the king, James I, respectively.
The Lord Chamberlain who was the patron for Shakespeare's acting company was Henry Carey, the 1st Baron Hunsdon. He served as Lord Chamberlain from 1585 until his death in 1596 and supported the Lord Chamberlain's Men, the acting company to which Shakespeare belonged. Under his patronage, the company flourished and performed many of Shakespeare's plays.
The Lord Chamberlain's Men
The underlined words "The Lord Chamberlain's Men" are an example of a proper noun. Proper nouns are specific names for particular people, places, organizations, or entities and are typically capitalized. In this case, it refers to a notable acting company in which Shakespeare was a key member.