Many way to implement this:
Older stuff uses a bunch of filters, each one made for the specific RBW, and switching them into the signal path. Especially the small RBW (e.g. 30Hz) filters are difficult, they operate at the third IF (usually in the 10 MHz range, usually you'd require a crystal filter made from a bunch of selected and thoroughly tuned crystals here)
In one particular case I've seen a cascaded crystal filter, where the Q of each stage could be varied by some electrical means (don't remember the details atm), resulting in an adjustable RBW
Less modern devices use a mixture of analog and digital filtering, changing the BW of a digital filter is rather simple by exchanging the set set of coefficients.
Modern devices most probably use some kind of digital processing for the RBW filter, and everything is hidden inside a DSP or FPGA chip.