Sharon Forbes and her daughter Savannah filmed the man, Alex Davies, at a National Action rally in Bath.
The video has been viewed thousands of times since they shared it.
Mr Davies subsequently told the BBC that he did not condone intimidation but had a right to say his views in public.
National Action is a far-right group that describes itself as fighting for a free white Britain.
Ms Forbes said she felt she had to confront Mr Davies, who was speaking at the rally: "I was so incensed and so angry at the views that he was spouting that I thought I just have to get them on video."
Since the video was posted on social media they have received abuse but Ms Forbes said the views of the people who had posted "are not going to intimidate us - not at all".
Savannah, who is mixed race, said: "We feel too strongly to be scared of people with such a small-minded view."
She added: "I didn't know that people like that still existed... and that people still had that mind set."
Mr Davies said he accepted his group had racist views and the rally was "low-key" until Ms Forbes began filming.

NHS Highland said a midwife-led community maternity unit (CMU) is to be in place at the hospital at the end of this month.
In November last year, the health board had anticipated introducing the change by April.
Maternity services are currently led by consultant obstetricians.
NHS Highland has said the majority of births at Caithness General do not require obstetrics.
The hopsital has no facilities for on-site specialist neonatal paediatric support or adult intensive care.
Complicated pregnancies are to be handled by "strengthened" services at Raigmore Hospital in Inverness and also close working with the ambulance service, the health board said.
It said it was introducing the CMU sooner than planned for staffing reasons.
Highland councillor Gillian Coghill has told BBC Alba that she was "surprised" by the health board's latest announcement on the change to the services.
She said a run-in period of about three months had been expected.

RWE Innogy UK had applied for a permit to divert water from the River Conwy to supply a new hydroelectric scheme near Betws-y-Coed.
Campaigners had feared the scheme would reduce the waterfall to a "trickle".
Natural Resources Wales refused permission saying it could harm the local environment.
RWE Innogy UK had said its Conwy Falls hydro power station plans, in Conwy Valley, could provide more than 3,200 households with power.
The plans, which had faced opposition from environmentalists, anglers, canoeists and businesses, would have seen a new hydro power scheme built 400 metres (1,312ft) upstream from the falls.
But on Thursday, NRW turned down the application for a permit to build and operate the plant saying the company wanted to abstract too much water at times from the river.
Sian Williams, head of operations north for NRW, said: "The Conwy is an important natural asset for the area and we've conducted a detailed assessment of the proposals before reaching this decision.
"We support sustainable developments in the right places.
"But the evidence we've scrutinised shows that the proposal submitted could harm the local environment. As such, we can't grant it a permit."
In March, Snowdonia National Park Authority's planning committee rejected the scheme.
RWE Innogy UK has been contacted for comment.
A mother and daughter claimed they have received online abuse after posting a video showing them confront a man about his racist comments.
[NEXT_CONCEPT]
Preparations are being made to change the operation of maternity services at Caithness General in Wick earlier than previously planned.
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Controversial plans for a £12m energy scheme at a waterfall beauty spot in Snowdonia National Park have been refused.