why are american ambassadors to Britain called ambassadors to the court of St. James

Hello poony-ga

Thank-you for your question.

The court of St James is the royal court of Britain.  This court is a
group of officials, including the monarch, who aid in governing the
country determining laws and regulations.  As ambassadors, in any
country not just the USA, are Britain's representatives in a country
they are aiding the court of St James.

Further information on the royal court of St James may be discovered here:
http://www.answers.com/topic/court-of-st-james-s
http://www.bartleby.com/59/13/courtofstjam.html
http://www.britainusa.com/sections/articles_show_nt1.asp?i=41129&L1=41127&L2=41129&a=28569&D=3
http://www.royal.gov.uk/output/Page589.asp

If you require any further information on this subject please do not
hesitate to ask for clarification and I will do my best to respond
swiftly.

If I may, I would express the last sentence in the answer differently:
As ambassadors for any country, not just the USA, they are accredited
by the court and recognized as the official representatives of their
countries.  They are the official channel of communication between the
British government and their country and are not subject to British
law (Diplomatic Rights).
The same applies, of course, for ambassadors in all other countries.