It is true that blood flows to wherever the brain is most activated and does fMRI measure the blood flow inside of the brain through oxygen content?
As a slight modification of your statement: blood flow increases wherever activity in the brain increases.
The type of fMRI that uses this principle is blood-oxygenation-level-dependent fMRI or BOLD fMRI. MRI in general detects signals by picking up proton signals from water molecules. This proton signal is basically caused by magnetizing the protons causing their spin to change. A subsequent powerful radiowave disrupts this spin and the following relaxation phase of the protons to the original state can be detected by MRI. Water, and hence protons are everywhere in the body, including the brain and the blood. Deoxygenated hemoglobin (hemoglobin without oxygen) in blood changes the proton signal in its immediate surroundings due to the magnetic properties of deoxyhemoglobin. This is caused by the fact that deoxygenated hemoglobin is paramagnetic and decreases the signal that protons release. In fact, it has been regarded as noise in structural MRI scans. Oxygenated hemoglobin does not have this property. Radiopaedia has a nice explanation as to exactly how the BOLD signal is used in BOLD fMRI, and I quote:
When a specific region of the cortex increases its activity in response to a task, the extraction fraction of oxygen from the local capillaries leads to an initial drop in oxygenated haemoglobin [...]. Following a lag of 2-6 seconds, cerebral blood flow (CBF) increases, delivering a surplus of oxygenated haemoglobin, washing away deoxyhemoglobin. It is this large rebound in local tissue oxygenation which is imaged.
So to sum up: brain activity increases the BOLD signal by picking up oxygen-changes after an increased blood flow to that specific part of the brain. So your statement that fMRI measures blood flow is technically incorrect. Doppler techniques can be used to measure the actual flow of blood.