At the heart of this misconception is the idea that if something happens naturally, then it can only happen naturally. But of course that's not true. Here are some examples.
Forest fires occur naturally. Does that mean that arson is a hoax?
People die of natural causes. But sadly, people are sometimes murdered. But if people can die on their own does that mean that murder does not happen?
It rains, which makes my lawn wet. But sometimes, a sprinkler is used to make the lawn wet. So the lawn can become wet for either natural or human-caused reasons.
Rivers have always flooded. But some floods are either caused by, or made worse by human actions. If a dam ruptures, the resulting flood is because of humans – not because floods happen on their own.
Many processes on Earth have more than one cause. The presence of a natural cause does not negate the reality of a human trigger.
Furthermore, the fact that climate has changed on its own gives us some very helpful information. Throughout geologic history, we know that more greenhouse gases in the atmosphere equates to a warmer climate, regardless of the source of the greenhouse gases (from the ocean, from wildfires, from volcanoes, from melting permafrost, etc.). So as humans add CO2 and other greenhouse gases to the atmosphere, we know that it also has the same warming effect. The physical process is the same either way.
Lastly, a key difference between past climate changes and today's climate change is the rate of change. Today's warming is much more rapid that past shifts in climate. This makes it harder for ecosystems to adapt.
In past geological ages the drivers of climate have been natural (volcanoes, plate tectonics, meteorites, cyanobacteria evolving and putting oxygen into our atmosphere, etc.), but now we as humans are the drivers of climate. This is why some scientists refer to today's geological era as the 'Anthropocene.' While it is scary to think that we as a species are responsible for altering the climate, it also highlights the fact that we humans are the first 'self-aware' climate driver and so could potentially do something about our actions.
Watch Katharine Hayhoe's lively explanation for this misconception:
This question relates to Climate Literacy Principle 4: Climate varies over space and time through both natural and man-made processes.
Even though this is a scientific question, sometimes the communication is more challenging than the science. See teaching materials about climate communication.