The other answers have correctly identified the character (It was Small Jon Umber, Lord of Last hearth) in question but none of them have dealt with the Other part of the Question.
Did they have an exit strategy?
No they did not have an exit strategy. They were there to finish the show.
Original Plan of Starks
Original plan of Starks was explained in Jon's War council:
Jon: We're digging trenches all along our flanks. They won't be able
to hit us the way that Stannis hit you[Wildlings], in a double
envelopment. A pincer move. They won't be able to hit us from the
Davos: It's crucial that we let them charge at us. They've got the
numbers, we need the patience. If we let him buckle our center, he'll
pursue. Then we'll have him surrounded on three sides.
Jon's vulnerability and Ramsay's plan
Jon however got emotional when Ramsay threw the bait of releasing Rickon. He pissed on the sound plan and rode head on, forcing his soldiers to abandon their fortified positions and follow Jon. You have to give Sansa this, she was not emotional. She knew Rickon was marked for death and had given him up for the dead. Jon however was still the boy Maester Aemon advised him to kill.
Ramsay knew about his weakness and planned to use it. He wanted to draw him out, envelop his army and destroy the Stark threat once and for all.
Battle turned sour as Starks were at complete disadvantage in head-on battle and eventually they were surrounded from three sides by Northmen and on their rear was the pile of corpses. Umbers specifically moved from there to start cutting down the broken Stark men with element of surprise.
Starks by then were finished and were just waiting to be cut down.
Hammer and Anvil
That's when Ramsay decided to deploy the Classic strategy of Hammer and Anvil. Ramsay wanted to draw Jon out of his crude fortifications so that he could use his numbers and horses for advantage. He used Rickon to draw Jon out and it worked, Jon moved into the trap like the idiot that he is. As this meme puts it:
Ultimately came the moment when Starks were surrounded by Boltons and pinned down against the corpse wall. Bolton infantry formed shield walls in three directions and started pushing the Starks back to their doom. The Anvil was ready.
Ramsay had counted on this, he had planned this. Then he turns to Lord Umber (Whom I won't be calling Small Jon out of respect for the books' character) and says this:
It's time, go.
That's when Lord Umber becomes the Hammer to crush the Starks between his forces and the Anvil of the surrounding Bolton infantry.
But they will get killed by spears of friendly Forces?
Correct, the risk of friendly casualties is always present in Hammer and Anvil tactics like all the other tactics. If there was no hammer, Starks would be left alone to try and break the shield wall. When Umber came, that's precisely what Starks were trying to do. Umber's attack made the Starks panic and had impact on their morale.
Difference in morals of fighting sides
It's not like Ramsay had any concern for loss of human life, soldiers were expendables.
Ser Davos forbade his soldiers to shoot arrows during the battle because that would have resulted in friendly fire:
Davos: We'll just kill our own men. Stand down.
But Ramsay had no such concerns and throughout the battle, Bolton arrows kept coming, falling on friend and foe alike.
But why would Umber risk his life? A Man who does nothing in battle risks his life as well. By attacking, he had a chance to quickly put an end to the battle. It was very likely that they would slaughter the Starks from the back long before Infantry managed to reach the Bridge, seeing as Stark men were broken.