In this case the original audio recording had some background noise, so I did a noise reduction in Goldwave, then a bandpass, then went through the recording by hand and replaced the gaps between the bursts of song with silence before I did the transcription.
The musician wren recording also gets quite polyphonic later on - birds are able to sing two distinct pitches simultaneously, and it seems that the musician wren is doing so at times in this recording.
For now I've just done a transcription of the first phrase, which is basically monophonic. One may be able to handle the polyphony by using bandpass to isolate the various parts - easier with birdsong which is close to a sine wave with few harmonics.
The aim is to remove the noise entirely, rather than to make an audio file that sounds natural - so there is no need for subtle parametric equalization - you can use a sharp cut off type lowpass, highpass and bandpass, combined with noise reduction. and repeat the process if necessary combining several methods.
This can get rid of the traffic noise and other lower or higher pitches almost entirely.
Then if the recording still has a fair bit of residual noise in between the notes of the song (e.g. distant bird song) then go through the recording setting it to silence by hand between every burst of song from the bird you want to transcribe.
With the musician wren, I set it to silence at the start and end of the recording, between the two bursts of song, and also for the gap between the first two phrases of the first very short burst of song, and a short gap that occurred about half way through the main burst of song. It doesn't take long to do and makes a big difference.
The trickiest sounds to deal with are the ones similar in pitch to the ones you want to transcribe - e.g. songs of other birds. So try to choose or make recordings with no other birds singing or only ones quieter - more distant - and far away in pitch from the one you want to transcribe.
If fairly quiet, then they are mainly a problem during the silent parts of the bird you want to transcribe, so that can be dealt with by setting the gaps in the recording to silence by hand. During the song they may add extra notes, especially if they are temporarily louder than the one you are transcribing, there you may be able to remove the extra notes by using a bandpass for the note you want to transcribe, for just that short section of the recording, if they are sufficiently different in pitch from the note you want to transcribe.
As for the tool to use for this - I use Goldwave and recommend it. For a free solution you can try Audacity. I'm sure there are many other possibilities.