To expand on Dan's answer, a ground offensive or attack involves moving the attacker's troops up to the enemy positions and engaging in combat with them. If the defender attempts to disengage from combat and retreat, the attacker can choose to (a) also disengage and move elsewhere, (b) occupy the positions that the defender has now abandoned or (c) pursue the defender's troops as he retreats, thus keeping the defender engaged in combat. Option (c) is the pursuit of the defender and prevention of disengagement that is called "to press the offensive". In a more metaphorical sense, "to press the offensive" is to follow up one offensive maneuver (such as a ground assault, an air assault or even just a punch or kick) with another. It means "to remain on the offensive and continue attacking the defender".

An "offensive" is a military attack. It is the opposite "defense." The Ukrainians are attacking and moving in. The Russians and separatists are trying to stop them. To "press" means to push, or to try hard at, or work hard at. "To press" means the Ukrainians are trying hard. They are heavily committed.