The upper stage gets left behind in Earth orbit. The Briz-M stage provides the kick to propel the payload to a higher orbit and then usually there's a kick motor that follows up, in this case, delivering a Trans-Mars Injection maneuver to set the spacecraft on its way to Mars.

EDIT: Actually realised I'm wrong since the Briz-M is the 4th stage, not the 3rd. The 3rd stage remains in orbit and then splashes down and the 4th stage actually provides the entire kick for TMI [1]

[1] http://blogs.esa.int/rocketscience/2016/03/13/why-exomars-ri...

Saw some diagram and it seems after the kick it has an additional burn to get it into a graveyard orbit instead of goong to mars along the payload. Maybe that burn which failed

The blog post I linked to in my previous comment suggests differently. Do you happen to know where you saw that diagram? Since I'm working on space debris, I'm curious what the target graveyard orbit is if it's not escaping.

Not an actual flight path http://www.russianspaceweb.com/images/spacecraft/planetary/m...