The term you are looking for is neurovascular coupling [1]. The increase in blood flow allows more oxygenated hemoglobin to flow through the capillary bed where the neuronal activity is taking place. This increases oxygen availability, which is necessary to maintain homeostasis and replenish the membrane potentials of the neurons, given that the brain has (essentially) no energy storage. There are multiple signalling pathways that control the mechanism, many of which are still poorly understood - for a review see [2].

I should add that every full moon a study comes out criticizing fMRI or drawing attention to its limitations, but researchers are very well aware of that. The fMRI response* to a task is neither temporally nor spatially specific, and it is confounded by anything that could remotely influence blood flow and blood volume. Figuring out how to deal with these issues is an active area of research - see an informal discussion here [3].

It's really a shame that the field is purported so negatively by the press, given it's the only thing we have to study brain function in humans with sub-millimeter spatial resolution non-invasively in-vivo.

[1] http://scholarpedia.org/article/Neurovascular_coupling [2] https://www.cell.com/neuron/pdf/S0896-6273(17)30652-9.pdf [3] https://practicalfmri.blogspot.com/2017/08/fluctuations-and-...

* Assuming the conventional Blood Oxygenated Level Dependent contrast used for functional imaging.