I'm not an expert but knowing there are usually two sides to every story I have been listening to a few Egyptians trying to explain what's going on. I'll try to give some of the alternative arguments I've heard - I'm not endorsing any of them.
But their candidates were elected by the people of Egypt.
The argument against this is that Muslims were put under considerable pressure in the Mosques to vote for the Muslim Brotherhood.
As such many Egyptian liberals/secularists consider the election was not free and fair - because politics was not separated from religion.
Post election the actions of Morsi to increase his powers and the violent tactics of the MB in persecuting other political and religious minorities e.g. anti-feminist activities and escalation of ethnic cleansing of Coptic Christians http://en.wikipedia.org/wiki/Persecution_of_Copts
did not promote a sense of democracy for all Egyptians. There were also ongoing faction fights between Islamist groups within the government making things chaotic.
And then removed from leadership by a military coup.
Apparently, the military only stepped in after many liberal/secularist Egyptians opposed to MB conducted huge ongoing protests. Morsi defied an ultimatum from the military to accommodate some of the demands of liberal/secular protestors. That precipitated the "coup".
And then folks protesting the coup started being slaughtered in the streets.
These MB protesters were organised from the Mosques (to include women and children) but were coordinated with other violent activities including burning of Coptic churches and attacks on government facilities and police stations. So further inflaming the liberals/secularists opposed to the religious Islamist politics by the MB.
I don't like to say in, but in this case, I think that the Muslim Brotherhood have the moral high ground.
The current military government is saying the MB have been behaving as "terrorists" against their political opposition for some time.
My own view on all of this: Recent massacres of MB protesters are certainly a tragic over-reaction from the military together with the angry secularist crowds. But the MB were also pushing their weight around first, often violently, so aren't blameless in the situation. So the whole saga may be a necessary process (by middle east standards) to purge Egypt of the ideals of the MB to run the country with Islamist ideas. I suppose we'll only be able to judge this if the military take Egypt to another democratic election within a reasonable timeframe and without the secularists having to start protesting against the military.