Because Julius Caesar died, what will happen to the Republic?
Although the conspirators say they're trying to save the Republic from Caesar's attempts to become king, they really don't end up saving it at all.
After Caesar is killed, the empire gets divided up between the members of the second triumvirate, Octavian, Antony and Lepidus.
These guys jockey with one another for power until finally (not seen in this play) Octavian defeats the forces of Antony and Cleopatra. This brings on the start of the Roman Empire.
So, the conspirators say they're trying to save the Republic from monarchy. But the end effect of their plot is that the last traces of the Republic die and Rome becomes an empire.
The death of Julius Caesar leads to the following:
1. Octavian takes it upon himself to raise an army to avenge Caesar.
2. The conspirators are killed and the hopes of the republic can be seen to have died with them (at least in hindsight).
3. There is power vacuum, which leads to another civil war. Octavian vs. Antony.
4. Antony is finally defeated at Actium by Octavian's right hand general, Agrippa in 31 B.C.
5. Octavian through political cunning is slowly able to establish the basis for an empire.
6. In short, the death of Caesar leads to a change in government.