As pointed by Wayne your provider box will convert the incoming signal to a video signal for you however, the OTA (over the air) signal will not work anymore on your CRT TVs once they stop sending the signal in the analogue form.
Digital and analogue are 2 different technologies: They both use carrier frequencies (your various channels) which are efficient for both technologies to transmit the signal to your home via air transmission, cable satellite etc... However, if your set has an analogue RF tuner, you will need to buy a converter which will change the incoming signal into an analogue signal in order for your old set to be able to display the video properly.
The property of the analogue technology is keying the carrier frequency in order to paint an image on your screen. In the case of digital, you are keying the level of the carrier with 0s and 1s which mean voltage or no voltage (typically 0 volt and 5 volts) in order to paint the image. Therefore, in order for an analogue CRT TV to recognise digital video, the digital video must be converted to analogue via the appropriate converter box.
Digital signal isn't always use for HD but you still get an improvement over analogue. My sister has been using digita
Do you remember when pay TV service was first introduced you needed a converter to convert TV sets from a tuner that could accept 13 channels to accept 36 channnels (I think that is what it was when it first started). So, some of the sort has to happen with the older analague technology in order for it to accept digital signals.
The only CRTs that I know does accept digital is when talking about HD RPTV as rear projection uses 3 CRTs, one for each color and if you have noticed, they are too fading away, not to many stores carry them and not too many manufacturers makes them anymore. Take a look at the attached document to have an overview of analogue and digital signals