As with all Roman politics, it was a mix of personal ambition and noble aims. It is extremely tough to tell how much was one and how much the other.
To give a full answer, we have to go back in time. Early in Rome's history it was a kingdom. The last king, Tarquinius Superbus was so cruel and evil that it induced the Romans to expel him, demolish the system of monarchy and install a Republic. Note that it wasnt really what many people think of as 'becoming democratic', the Republic was decidedly undemocratic in the way we currently understand democracy, but it was more about moving away from one man being in absolute control towards sharing the power amongst many elites. Simply put, it was decided that the ambition of these different elites would serve as a check/balance against one particular man getting too powerful and thus making Rome vulnerable to the sort of sick tyranny that happened under its Kings.
This cultural dynamic led to the Romans having a massive aversion to individuals becoming too powerful, to the extent that accusing someone of wanting a crown was one of the most lethal and disparaging accusations that could be levelled against a politician/general. It was the 'literally Hitler' equivalent in ancient Rome.
So of course, by the time Julius Caesar had declared himself dictator for life, you can imagine this was a problem for many. Interestingly, the general public didnt seem to mind as much as you might have expected, since Julius Caesar cleverly managed to portray himself as a hero of the people and his accomplishments gave the Romans a fair bit of pride and glory (and wealth). However the Romans that used to be the heads of the Republic were of course not happy since their own power had been diminished.
Enter Marcus Junius Brutus and the other conspirators. Now we get to answering your question.
Brutus was a powerful man in the Republic, but power was a finite resource in Rome. Julius Caesar had most of it, leaving very little for anyone else. Brutus (and the other conspirators) had only one hope of regaining their former prestige and power - getting rid of Caesar
There was certainly a 'noble' element to their thought process, in that there wqs very valid reasons for not trusting the tyranny of one man, as exemplified by Romans traditional mistrust of Kings. They may well have seen themselves as heroes of Rome, restoring traditional values and government in the face of this upstart Caesar.
To top it all off, Brutus was the descendant of Lucius Junius Brutus - the man who overthrew Tarquinius Superbus. His ancestors were the one who got rid of the king, and set up the Republic. In this was he had a family legacy of overthrowing tyrants, and this was certainly a factor. In Ancient Rome a powerful strategy was leveraging your family history, and in a way Brutus had the perfect opportunity to both take the power back, spinning a story of himself as carrying his great legacy of tyrant killing, and restore the Republic to boot. It provided a very strong nucleus and justification for the assassination plot to form around.
EDIT: In summary the motives were a mix of 1 and 2, with 3 being an additional factor that helped the plot to form and gather momentum
EDIT 2: Some others have presented a much better view of the issues in replies to this. Apologies for any oversimplifications, I'd delete this post if not for the great replies it's had. If the users would post their views as top level comments I'll happily delete this
There were a few major reasons and a large number of minor ones. The major ones were that many of the conspirators were aristocrats who disagreed with all the power Caesar had won for himself. They thought that what he was doing was stifling the principles of the republic.
They believed that laws should be made through the debate of Romes leading magistrates and leading aristocratic families. They also wanted to return to having the hope that they could climb to the highest rung of the political ladder. With one man in charge, that was obviously much more difficult.
Caesar actually made many great decisions during his dictatorship, but it was the fact that he was making them behind closed doors that was vexing many.
Other reasons were that Caesar was rumored to want to make himself king, which would have meant that the ambitions of many leading citizens would be squashed. Sulla had been named dictator for life before, but he laid down his power (something Caesar thought was political ineptitude). But Caesar had defeated all of his enemies and showed no signs of giving up powers as many senators hoped he would.
On top of all that he was about to set out on a three year campaign against Parthia to avenge Crassus' defeat some 10 years earlier. Had it been successful, it would have added immense personal glory to his prestige and all but assured the people would be ok with him establishing a monarchy. It's worth noting that, even though the common people might have disagreed with him from time to time, they adored him and rioted at his funeral.
He was also extending citizenship to a number of established provincial families, which expanded the senate. While this wasn't meant to dilute the power of the individual senator, it was meant to basically make all the provincial leaders his clients something that further frustrated the conspirators.
In the end, the conspirators did believe it was the right thing to do for the republic, but their concept of the republic was a system that favored the aristocracy over the greater good and tradition over actually getting things done. They killed Caesar because they thought it would best serve the continual interests of the aristocracy.
Sources: "Ceasar: Life as a Colossus" "Rubicon: The Last Years of the Roman Republic"