The Conservative MP was stabbed multiple times during a meeting with his constituents in Essex on Friday.
A 25-year-old British man is being held under the Terrorism Act.
In a statement, his family said they were trying to understand "why this awful thing has occurred...nobody should die in that way. Nobody".
Sir David, 69, was married with four daughters and a son.
The family said the "wonderful" tributes paid to him by friends, constituents and the public had given them strength.
"We have realised from tributes paid that there was far, far more to David than even we, those closest to him, knew," they added.
"We are enormously proud of him. Our hearts are shattered."
Raised as a Roman Catholic, Sir David was known politically as a social conservative and a prominent campaigner against abortion.
He was also a committed campaigner on animal welfare issues, and supported a ban on fox hunting.
A Conservative MP since 1983 - first in Basildon and, from 1997, in Southend West - he was a champion for the town he represented, particularly in his long-running campaign to make Southend a city.
His family have asked people to support campaigns that he was involved in, including fundraising for a memorial to Dame Vera Lynn, who he thought "epitomised the strength of the nation" - and to help Southend gain city status.
They described Sir David as strong and courageous, a patriot and a man of peace.
"We ask people to set aside their differences and show kindness and love to all. Please let some good come from this tragedy.
"We are absolutely broken, but we will survive and carry on for the sake of a wonderful and inspiring man."
Tributes to Sir David have been pouring in from politicians and constituents, with Home Secretary Priti Patel saying his "infectious personality" meant he "touched so many lives".
Over the weekend, people gathered for candlelit vigils in Leigh-on-Sea to mark Sir David's life and attended a church service to share their memories of him.
Many constituents have reflected on his gentle nature and willingness to listen and to help.
On Monday afternoon, MPs will pay tribute to their late colleague in the House of Commons ahead of a church service in his memory at Westminster Abbey.
Detectives are continuing to hold the 25-year-old man at a London police station and have until Friday to question him.
Whitehall officials confirmed the man's name as Ali Harbi Ali, and said he was a British man of Somali heritage.
The BBC understands he was referred to the counter-terrorist Prevent scheme some years ago, but was never a formal subject of interest to MI5.
It also understands that his father, Harbi Ali Kullane, who was previously an adviser to Somalia's prime minister, has been visited by police who have taken his phone for analysis.
Police officers have spent the weekend searching three addresses in London.
It is thought a converted Victorian property in Lady Somerset Road in north-west London is linked to the investigation. Neighbours said officers started searching it late on Friday night.
Further searches, also believed to be part of the inquiry, have been taking place at a property in Bounds Green Road, north London, and another in Cranmer Road, Croydon.