AudioSource in Update

First thing is to make sure that your AudioSource component is not set to "Play on Awake" secondly, Play() will try to start the audiosources clip, so calling it each frame as you are here would keep restarting it over and over, which is likely why you only hear it play once switchedOn is false and you stop calling Play. Try this.

 using UnityEngine;
 using System.Collections;
 
 public class Radio : MonoBehaviour
 {
     public AudioSource speaker;
     public bool switchedOn;
     
     void Update()
     {
         if (switchedOn == true && speaker.isPlaying == false)
         {
             speaker.Play();
         }
         else if (switchedOn == false && speaker.isPlaying == true)
         {
             speaker.Stop();
         }
     }
 }