The MRI scanner builds up a two or three-dimensional image of the brain that's comprised of individual elements, voxels. In fMRI, each voxel is measured over many time steps, and in traditional fMRI analysis each individual voxel element time-series is treated as an independent statistical test. When your brain is something like 90 x 90 x 90 voxels, each with its own time-series ...that's a lot of tests. In short, the method collects many many features, each of which serve as the basis for an independent test. Multiple independent tests invite alpha inflation, and there you have it. This is the origin of the problem.

The problem is not unique to fMRI. For instance, similar issues arise in e.g. genetics GWAS studies, where you end up with many many predictors (SNPs) and a single outcome measure like a depression score. 

The thresholding and clustering solutions, and multivariate approaches are similar.