Cook, 32, quit as skipper after four-and-a-half years and a record 59 matches in charge.
"He was getting drained by the relentlessness of being England captain," Strauss told the BBC.
England lost a test series 4-0 in India at the end of last year, prompting Cook to stand down.
"We know it has been a tough winter and it was an obvious time for him to step back and reflect and consider and have thoughts about what was right for the team moving forwards," Strauss said.
"In my conversations with him in January it became clear that Alastair felt a huge amount of energy, drive and determination was needed to drive the team forward over the next 12 months."
Cook, England's leading run-scorer in test cricket, led the side to two Ashes victories but the team under-performed in 2016 during which they lost eight test matches.
"You are the only one who knows how much gas you have left in the tank and how much the many demands of being England captain are taking out of you," said Strauss, Cook's former opening partner.
"He feels it is time for new blood, new impetus and fresh thinking and allowing someone else to take over and do that."
Joe Root is strong favourite to replace Cook at the helm but Strauss said he was not ruling anyone out of the running ahead of home series against South Africa and West Indies this year before the next Ashes contest in Australia.
(Reporting by Ed Osmond; Editing by Clare Fallon)
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