The lyrics are about the resurrection of Jesus Christ.
Our God is alive
Our God, Jesus, is not dead, even though he was killed, but is alive because he rose from the dead.
The tomb with soldiers watched in vain
After bring crucified, the body of Jesus was placed in a tomb and soldiers were placed on guard at it. However Jesus did arise, and the soldiers were unable to prevent it.
was borrowed for three days.
Jesus was laid in a tomb, but after three days rose again. The tomb was not needed after that - it was only borrowed, not kept permanently.
His body there would not remain
our God has robbed the grave
God, like a grave robber, has taken the body out of the grave - not by stealing it, but because he has the power to do so. Moreover he does the same for all of us, in that he will resurrect us all in due time - 'cheating' the enemy of the punishment that we deserve for our sins, which is death - the ultimate 'robbing' of the ultimate grave.
It is possible to get into trouble by taking that last line as too literal and exact a theological statement. God is not being deceitful or dishonest in 'robbing' the grave. It's a song lyric, not a doctrinal statement.