You have to think of "es que" as a unit that you can put in front of any sentence. It's used to emphasise that the sentence is a reason or explanation for whatever was asked or said first. You can translate it as "because", although it feels more like a filler word than "porque". Compare the following sentences:
Es que te quiero.
Soy de Madrid.
Es que soy de Madrid.
The second versions imply that an explanation is being given, for example after a lover is surprised at an unexpected gift or after somebody expresses surprise at your familiarity with Madrid street names.
In the quoted song, we could simply say "Tú sin mí y yo sin ti, dime quién puede ser feliz". Putting "es que" in front is just an example of this common pattern in action.
As mentioned in one of the answers, this use is often frowned upon, at least here in Spain. While not grammatically wrong, it is true that it has always been a pet peeve of teachers because children tend to overuse it and start virtually every sentence with "es que". For example, a child raising his hand at a primary school could say "Es que necesito hacer pis". The "es que" indicates that what follows is the justification of why they were raising their hand, but it's often used even when stating facts, as in "Es que los romanos conquistaron Hispania" or "Es que el agua se congela a 0 grados". Hence the common feeling that it is a useless, bad-style filler phrase.