"There was rain up to about 3,500 metres (11,483 feet) and perhaps a little fresh snow (at the 4,010 metre summit), so they probably slipped because there was ice underneath the snow," Arthur Anthamatten told AFP.
Speaking from Weissmies Hut at 2,726 metres (8,944 feet), Anthamatten said he had spoken to the party on Monday night over dinner at the refuge.
Their plan was to leave the refuge very early the next day after breakfast, at about 4.30am, he said.
"We didn't have many climbers on Monday night when they came to stay with us. We spoke about the refuge, they said they wanted to climb the mountain the next day," said Anthamatten.
"One of my staff talked to the girl who said she had already climbed some mountains and she enjoyed getting to the top," Anthamatten added.
"They said they had already done some climbing in Switzerland; three years ago they came to Saas Tal (in Valais) and they weren't afraid," said Anthamatten, who added that he and other staff at the refuge had been shocked by the scale of the accident on the relatively straightforward climb.
"This kind of thing can happen, but never five people killed in one go. Two or three people perhaps, but this is not a difficult mountain," said Anthamatten, whose family has run the refuge since the 1920s.
Police confirmed on Wednesday that the climbers were all German and that the accident happened at about 1.30pm on Tuesday.
Five members of the climbing party slipped hundreds of feet to their deaths on their way down from the summit, the statement said, adding that the climbers were not roped together when they fell.
Of the five victims, police confirmed that two were children aged 14 and 20. Their father -- the sixth member of the group -- had been too unwell to climb the final 100 metres up to the summit and stayed behind while the others went on.
A man aged 44 also died in the fall, along with his 17-year-old son and a 21-year-old friend, the police statement said, adding that investigations are continuing into the "tragic accident".