On the recording you will also hear a piano playing a chord and then a note before the count-in to the song begins. This is to establish the sound of the key in your mind and give you the starting pitch before you begin singing.
In this case the chord is a C chord, which establishes the sound for the key of C major. The note following the chord is a G note, which is the starting note of the song. As you hear the C chord, think do in your mind, then as you hear the G note, think so. This process is used with all the songs on the recording.
When you sing with other musicians, ask one of them to play both the chord which establishes the key and the starting note before you count the song in. If you ask for a cue note they will know what you mean. This way you will always be confident of starting on the correct pitch and will easily hear its relationship to the key note (do).