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The Guardian reports that the UK fracking tsar, Natascha Engel, has resigned and blamed anti-fracking activism for "fear-mongering", saying:

“There is much to be optimistic about how developing technologies – including fracking – can help us accelerate the reduction in CO2 and grow our economy. Sadly today only those who shout get heard.”

To me this seems completely illogical. I would think that fracking can only increase the supply of fossil fuel, that any increase in supply causes some increase in consumption, and that any increase in consumption of fossil fuel causes an increase in CO2.

Is there any validity to her claim?

First, let's agree on definitions. Fracking (or hydraulic fracturing) is a process by which fluids are drawn from the ground. In this context, the fluid would either be oil or natural gas.

From the article that you link:

Engel’s resignation letter said: “The UK is currently spending £7bn a year on importing gas – money that is not being used to build schools, hospitals or fix the potholes in our roads. Developing our own shale gas industry would mean money going into the Treasury rather than out.”

She added: “We know shale gas can be extracted safely. [...]

So she is discussing shale gas which is simply natural gas trapped in shale.

Also from the article:

Fracking, she said, had the potential to create jobs, economic security and provide a cleaner alternative to coal and biomass.

So she is specifically talking about substituting natural gas for coal. Some sources that support natural gas being better than coal in terms of greenhouse emissions:

• Natural Gas Really Is Better Than Coal


• Environmental Impacts of Natural Gas

Note that natural gas itself is a worse greenhouse gas than carbon dioxide. So to be cleaner than coal, they have to be careful not to let it leak.

It also may be worth noting that coal is a base load power, meaning that the plant is started and then run continuously. Turning the plant on and off is a lengthy process and not something that they do in response to variation in demand during the day. Coal power plants have a thermal mass, meaning that they continue producing electricity even after the flame is smothered.

Natural gas is on demand power. The generators can be smaller and turned on to meet demand. As such, natural gas is a more natural method to use with renewables like solar and wind than coal is. This is because those renewables are on supply power. They provide power when the sun is shining or the wind is blowing. But if you want power on a calm night, they don't help you.