Compressed sensing is a term from the field of digital signal processing. When a signal is digitally sampled, like it happens in an MRI scanner, the signal is not recorded continuously (like old cassette players used to do) but at intervals. A famous theorem from digital signal analysis, the Nyquist theorem, states that for constructing a perfect MR image of 256 x 256 pixels, it is required to sample 256 lines in k-space, each sampled in 256 positions. By doing less, the acquisition will be faster, but the reconstructed image will always be distorted one way or another.
This is exactly what happens with traditional acceleration techniques in MRI, such as halfscan, radial, spiral, increased voxel size and parallel imaging. All of these methods skip parts of k-space during acquisition in order to reduce acquisition time.