Liverpool lore has it that the switch to an all-red kit came in a European Cup game against Anderlect on November 25, 1964.
According to the recollections of Ian St John, in the lead-up to that game, manager Bill Shankly asked Ron Yeats to model a pair of red shorts with the classic shirt.
He thought the colour scheme would carry psychological impact — red for danger, red for power. He came into the dressing room one day and threw a pair of red shorts to Ronnie Yeats.
‘Get into those shorts and let’s see how you look,’ he said.
‘Christ, Ronnie, you look awesome, terrifying. You look 7ft tall.’
St John then had an additional idea:
‘Why not go the whole hog, boss?’ I suggested.
‘Why not wear red socks? Let’s go out all in red.’ Shankly approved and an iconic kit was born.
Liverpool beat the Belgians 3-0 at Anfield, but it took another while for St John’s suggestion to be taken on board. Look, for example, at the pictures used by the Daily Mail in an article about the 50th anniversary of the all-red strip, back in November 2014:

They have helpfully coloured in the top image but that and the one below it clearly show the socks to be white with red tops – the shorts appear lighter due to being made of a different material to the shirt.
Thankfully, David Prentice of the Liverpool Echo is an old-fashioned journalist in that he trusts his own eyes rather than slavishly repeating stories passed down and archive research by him revealed further layers to the story.

Liverpool had begun the 1964-65 season in the kit in which they had won the league, featured white shorts and socks with red trim.
A notable change was made against Anderlecht, as noted by Echo journalist Leslie Edwards, but the white socks remained:

Liverpool, with vivid red shorts which even goalkeeper Tommy Lawrence wore, looked strange.
As we saw in a recent article about the return of white sleeves to Arsenal’s kit in 1967, the Football League didn’t allow kits to be altered mid-season, so Liverpool were back in white shorts for the visit of Tottenham Hotspur that weekend.

In fact, it wasn’t until the away leg against Anderlecht, a 1-0 win, that the new red shorts were joined by red socks.
That was to remain the European kit and it was also a look that the Reds used in the FA Cup that season, with all-red worn at Anfield for the first time in the sixth-round replay against Leicester City on March 10, 1965.

Liverpool’s European Cup journey – including a quarter-final coin-toss victory over Köln after a pair of scoreless draws a 2-2 replay – ended with defeat to Inter Milan in the semi-finals, but they did make it to the FA Cup final.
With the colours of the crest reversed and a ‘Wembley 1965’ inscription added, they beat Leeds United 2-1 after extra time to claim a first trophy in all-red. Quite a few more would follow.