It's like saying:
I get headaches most days after work. Tonight I came home and the headaches started up again.
The fact that there are headaches is established in the first sentence as a general thing. Then, the definite article can be used to refer to them in the second sentence.
This is the fourth line from the Queen song:
(Take a look at yourself in the mirror and cry)
Take a look in the mirror and cry
It establishes a sense of somebody who cries. Therefore, there are tears. These are not a single instance of tears, but tears in general.
By the time we get to the lyric in question, it's no surprise that there is crying (and more tears):
I work hard (he works hard) every day of my life
I work 'til I ache my bones
At the end (at the end of the day)
I take home my hard-earned pay all on my own (goes home on his own)
I get down (down) on my knees (knees)
And I start to pray (praise the Lord)
'Til the tears run down from my eyes, Lord
Just as we can say that the headaches start up again, this person is praying and crying until the tears start up again. The pain is back. The tears are back.
In some cases, however, the definite article can be used without ever having established anything beforehand:
The drinking water from my taps doesn't taste that bad.
The drinking water is not something particular (it's not this glass of water as opposed to that glass of water)—but it is something identified in context. It's the drinking water from my taps.
Similarly, in the song, it's the tears that happen when the person is home, feeling sad, and crying.
It's not a specific object, but an identified idea that has an ongoing existence with multiple instances.