Multiple independent lines of scientific evidence demonstrate that people are causing climate to change.
The growing chemical signature of carbon from fossil fuels in the atmosphere (and basic math) demonstrate that people are causing carbon dioxide concentrations to increase.
The warming influence of greenhouse gases (e.g., carbon dioxide) is clear from laboratory experiments, evidence from past changes in climate due to greenhouse gases, and the role of greenhouse gases on other planets (e.g., Venus is much hotter than Mercury despite being further from the sun).
Additional lines of evidence relate to the patterns of climate change underway. These patterns match the characteristics expected from greenhouse gases well and do not match the characteristics we would expect from the other factors that could change climate such as the sun, volcanoes, aerosols, land-use patterns, or natural variability. It is possible that these other factors could have contributed a small net warming or net cooling influence on the climate system recently. However, this is in addition to the human contribution to climate to change and small by comparison.
Think of it like a whodunit where the list of suspects is the potential causes of climate change. Critically, each suspect has a unique fingerprint. Scientists have worked hard to identify the potential causes of climate change and the patterns of change they would produce. The changes in climate that we’ve witnessed over the last several decades match the fingerprint of greenhouse gases well and do not match what we would expect to see from the usual suspects: the sun, volcanoes, aerosols, land-use patterns, or natural variability. The combination of fingerprint analysis with what we know about greenhouse gases is conclusive evidence that humans are causing climate to change.
How we know climate is changing
Why the claim that "People are contributing to climate change" is accurate but misleading.
References for those who want to dig deeper: Stocker et al. 2013, NAS, AMS Climate Change statement, etc.