When Maxwell added his displacement current term to what we now call Maxwell's equations, he realized that the equations now had solutions with waves propagating in free space. Since the electric constant and magnetic constant had been separately measured (what we today call ε0 and μ0). He was able to calculate the speed of these waves and found that their speed was very close to the speed of light that had been measured by Fizeau. He then hypothesized that light was an electromagnetic disturbance. He wrote,
"we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena"
The existence of electromagnetic waves was later verified by Heinrich Hertz.
I think light oscillates too fast to measure the electric and magnetic fields directly, but we can infer them through their interaction with charged particles.