Extending one's sympathy is letting another person know how sad you feel about what happened to him. Example: "I would like to extend my sympathy for the recent death of your mother." Reply: "I appreciate that very much."

"Sympathy" is feeling bad when another person has had a loss, and extending your sympathy is letting that person know how bad you feel for their loss. Example: "We would like to extend our sympathy for the loss of your cat."

"Extending your sympathy" is a polite way to acknowledge in words that another person has lost something or someone important. Example: "I want to extend my sympathy to you about your mother." Reply: "Thank you."